Data.Colour.Matrix:determinant from colour-2.3.3, A

Percentage Accurate: 73.5% → 82.3%

Time: 6.5s

Alternatives: 21

Speedup: 0.5×

• 57.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (+
  (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i))))
  (* j (- (* c a) (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)))

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(t * i)))) + Float64(j * Float64(Float64(c * a) - Float64(y * i))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 73.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (+
  (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i))))
  (* j (- (* c a) (* y i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)))
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)))

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(t * i)))) + Float64(j * Float64(Float64(c * a) - Float64(y * i))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 82.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{if}\;\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + t\_1 \leq \infty:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(j \cdot a - b \cdot z\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* c a) (* y i)))))
   (if (<=
        (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) t_1)
        INFINITY)
     (+ (fma (- (* z y) (* a t)) x (* (- b) (- (* c z) (* i t)))) t_1)
     (* (- (* j a) (* b z)) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = j * ((c * a) - (y * i));
	double tmp;
	if ((((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + t_1) <= ((double) INFINITY)) {
		tmp = fma(((z * y) - (a * t)), x, (-b * ((c * z) - (i * t)))) + t_1;
	} else {
		tmp = ((j * a) - (b * z)) * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	tmp = 0.0
	if (Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(t * i)))) + t_1) <= Inf)
		tmp = Float64(fma(Float64(Float64(z * y) - Float64(a * t)), x, Float64(Float64(-b) * Float64(Float64(c * z) - Float64(i * t)))) + t_1);
	else
		tmp = Float64(Float64(Float64(j * a) - Float64(b * z)) * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], Infinity], N[(N[(N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision] * x + N[((-b) * N[(N[(c * z), $MachinePrecision] - N[(i * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + t$95$1), $MachinePrecision], N[(N[(N[(j * a), $MachinePrecision] - N[(b * z), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i)))) < +inf.0

   1. Initial program 91.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 91.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   if +inf.0 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i))))

   1. Initial program 0.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot j - b \cdot z\right) \cdot \color{blue}{c} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot j - b \cdot z\right) \cdot \color{blue}{c} \]

      3. lower--.f64 N/A

         \[\leadsto \left(a \cdot j - b \cdot z\right) \cdot c \]

      4. *-commutative N/A

         \[\leadsto \left(j \cdot a - b \cdot z\right) \cdot c \]

      5. lower-*.f64 N/A

         \[\leadsto \left(j \cdot a - b \cdot z\right) \cdot c \]

      6. lower-*.f64 44.9

         \[\leadsto \left(j \cdot a - b \cdot z\right) \cdot c \]

   4. Applied rewrites 44.9%

      \[\leadsto \color{blue}{\left(j \cdot a - b \cdot z\right) \cdot c} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 82.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{if}\;t\_1 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(j \cdot a - b \cdot z\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i))))
          (* j (- (* c a) (* y i))))))
   (if (<= t_1 INFINITY) t_1 (* (- (* j a) (* b z)) c))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
	double tmp;
	if (t_1 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = ((j * a) - (b * z)) * c;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
	double tmp;
	if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = ((j * a) - (b * z)) * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)))
	tmp = 0
	if t_1 <= math.inf:
		tmp = t_1
	else:
		tmp = ((j * a) - (b * z)) * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(t * i)))) + Float64(j * Float64(Float64(c * a) - Float64(y * i))))
	tmp = 0.0
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(j * a) - Float64(b * z)) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)));
	tmp = 0.0;
	if (t_1 <= Inf)
		tmp = t_1;
	else
		tmp = ((j * a) - (b * z)) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(t * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, Infinity], t$95$1, N[(N[(N[(j * a), $MachinePrecision] - N[(b * z), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i)))) < +inf.0

   1. Initial program 91.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   if +inf.0 < (+.f64 (-.f64 (*.f64 x (-.f64 (*.f64 y z) (*.f64 t a))) (*.f64 b (-.f64 (*.f64 c z) (*.f64 t i)))) (*.f64 j (-.f64 (*.f64 c a) (*.f64 y i))))

   1. Initial program 0.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot j - b \cdot z\right) \cdot \color{blue}{c} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot j - b \cdot z\right) \cdot \color{blue}{c} \]

      3. lower--.f64 N/A

         \[\leadsto \left(a \cdot j - b \cdot z\right) \cdot c \]

      4. *-commutative N/A

         \[\leadsto \left(j \cdot a - b \cdot z\right) \cdot c \]

      5. lower-*.f64 N/A

         \[\leadsto \left(j \cdot a - b \cdot z\right) \cdot c \]

      6. lower-*.f64 44.9

         \[\leadsto \left(j \cdot a - b \cdot z\right) \cdot c \]

   4. Applied rewrites 44.9%

      \[\leadsto \color{blue}{\left(j \cdot a - b \cdot z\right) \cdot c} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 64.6% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot z - i \cdot t\right) \cdot b\\ \mathbf{if}\;x \leq 2 \cdot 10^{+17}:\\ \;\;\;\;\left(c \cdot a - i \cdot y\right) \cdot j - t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y - a \cdot t\right) \cdot x - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (- (* c z) (* i t)) b)))
   (if (<= x 2e+17)
     (- (* (- (* c a) (* i y)) j) t_1)
     (- (* (- (* z y) (* a t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((c * z) - (i * t)) * b;
	double tmp;
	if (x <= 2e+17) {
		tmp = (((c * a) - (i * y)) * j) - t_1;
	} else {
		tmp = (((z * y) - (a * t)) * x) - t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((c * z) - (i * t)) * b
    if (x <= 2d+17) then
        tmp = (((c * a) - (i * y)) * j) - t_1
    else
        tmp = (((z * y) - (a * t)) * x) - t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((c * z) - (i * t)) * b;
	double tmp;
	if (x <= 2e+17) {
		tmp = (((c * a) - (i * y)) * j) - t_1;
	} else {
		tmp = (((z * y) - (a * t)) * x) - t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((c * z) - (i * t)) * b
	tmp = 0
	if x <= 2e+17:
		tmp = (((c * a) - (i * y)) * j) - t_1
	else:
		tmp = (((z * y) - (a * t)) * x) - t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(c * z) - Float64(i * t)) * b)
	tmp = 0.0
	if (x <= 2e+17)
		tmp = Float64(Float64(Float64(Float64(c * a) - Float64(i * y)) * j) - t_1);
	else
		tmp = Float64(Float64(Float64(Float64(z * y) - Float64(a * t)) * x) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((c * z) - (i * t)) * b;
	tmp = 0.0;
	if (x <= 2e+17)
		tmp = (((c * a) - (i * y)) * j) - t_1;
	else
		tmp = (((z * y) - (a * t)) * x) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(c * z), $MachinePrecision] - N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[x, 2e+17], N[(N[(N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision] - t$95$1), $MachinePrecision], N[(N[(N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2e17

   1. Initial program 73.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) - b \cdot \left(c \cdot z - i \cdot t\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto j \cdot \left(a \cdot c - i \cdot y\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot t\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot j - \color{blue}{b} \cdot \left(c \cdot z - i \cdot t\right) \]

      3. *-commutative N/A

         \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - b \cdot \left(c \cdot z - i \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j - b \cdot \left(c \cdot z - i \cdot t\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j - \color{blue}{b} \cdot \left(c \cdot z - i \cdot t\right) \]

      6. lift--.f64 N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j - b \cdot \left(c \cdot z - i \cdot t\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j - b \cdot \left(c \cdot z - i \cdot t\right) \]

      8. *-commutative N/A

         \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - b \cdot \left(c \cdot z - i \cdot t\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - b \cdot \left(c \cdot z - i \cdot t\right) \]

      10. *-commutative N/A

          \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - i \cdot t\right) \cdot \color{blue}{b} \]

      11. *-commutative N/A

          \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - t \cdot i\right) \cdot b \]

      12. lower-*.f64 N/A

          \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - t \cdot i\right) \cdot \color{blue}{b} \]

      13. lift--.f64 N/A

          \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - t \cdot i\right) \cdot b \]

      14. lift-*.f64 N/A

          \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - t \cdot i\right) \cdot b \]

      15. *-commutative N/A

          \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - i \cdot t\right) \cdot b \]

      16. lower-*.f64 63.8

          \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - i \cdot t\right) \cdot b \]

   4. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - i \cdot t\right) \cdot b} \]

   if 2e17 < x

   1. Initial program 73.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in j around 0

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) - b \cdot \left(c \cdot z - i \cdot t\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot t\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot z - a \cdot t\right) \cdot x - \color{blue}{b} \cdot \left(c \cdot z - i \cdot t\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x - \color{blue}{b} \cdot \left(c \cdot z - i \cdot t\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      6. *-commutative N/A

         \[\leadsto \left(z \cdot y - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(z \cdot y - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      8. *-commutative N/A

         \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      10. *-commutative N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot \color{blue}{b} \]

      11. *-commutative N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b \]

      12. lower-*.f64 N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot \color{blue}{b} \]

      13. lift--.f64 N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b \]

      14. lift-*.f64 N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b \]

      15. *-commutative N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot b \]

      16. lower-*.f64 67.3

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot b \]

   4. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 63.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{+17}:\\ \;\;\;\;\left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= x 2.2e+17)
   (- (* (- (* c a) (* i y)) j) (* (- (* c z) (* i t)) b))
   (- (* (- (* z y) (* a t)) x) (* (* c z) b))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= 2.2e+17) {
		tmp = (((c * a) - (i * y)) * j) - (((c * z) - (i * t)) * b);
	} else {
		tmp = (((z * y) - (a * t)) * x) - ((c * z) * b);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (x <= 2.2d+17) then
        tmp = (((c * a) - (i * y)) * j) - (((c * z) - (i * t)) * b)
    else
        tmp = (((z * y) - (a * t)) * x) - ((c * z) * b)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (x <= 2.2e+17) {
		tmp = (((c * a) - (i * y)) * j) - (((c * z) - (i * t)) * b);
	} else {
		tmp = (((z * y) - (a * t)) * x) - ((c * z) * b);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if x <= 2.2e+17:
		tmp = (((c * a) - (i * y)) * j) - (((c * z) - (i * t)) * b)
	else:
		tmp = (((z * y) - (a * t)) * x) - ((c * z) * b)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (x <= 2.2e+17)
		tmp = Float64(Float64(Float64(Float64(c * a) - Float64(i * y)) * j) - Float64(Float64(Float64(c * z) - Float64(i * t)) * b));
	else
		tmp = Float64(Float64(Float64(Float64(z * y) - Float64(a * t)) * x) - Float64(Float64(c * z) * b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (x <= 2.2e+17)
		tmp = (((c * a) - (i * y)) * j) - (((c * z) - (i * t)) * b);
	else
		tmp = (((z * y) - (a * t)) * x) - ((c * z) * b);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[x, 2.2e+17], N[(N[(N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision] - N[(N[(N[(c * z), $MachinePrecision] - N[(i * t), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - N[(N[(c * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2e17

   1. Initial program 73.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right) - b \cdot \left(c \cdot z - i \cdot t\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto j \cdot \left(a \cdot c - i \cdot y\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot t\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot j - \color{blue}{b} \cdot \left(c \cdot z - i \cdot t\right) \]

      3. *-commutative N/A

         \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - b \cdot \left(c \cdot z - i \cdot t\right) \]

      4. *-commutative N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j - b \cdot \left(c \cdot z - i \cdot t\right) \]

      5. lower-*.f64 N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j - \color{blue}{b} \cdot \left(c \cdot z - i \cdot t\right) \]

      6. lift--.f64 N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j - b \cdot \left(c \cdot z - i \cdot t\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j - b \cdot \left(c \cdot z - i \cdot t\right) \]

      8. *-commutative N/A

         \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - b \cdot \left(c \cdot z - i \cdot t\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - b \cdot \left(c \cdot z - i \cdot t\right) \]

      10. *-commutative N/A

          \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - i \cdot t\right) \cdot \color{blue}{b} \]

      11. *-commutative N/A

          \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - t \cdot i\right) \cdot b \]

      12. lower-*.f64 N/A

          \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - t \cdot i\right) \cdot \color{blue}{b} \]

      13. lift--.f64 N/A

          \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - t \cdot i\right) \cdot b \]

      14. lift-*.f64 N/A

          \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - t \cdot i\right) \cdot b \]

      15. *-commutative N/A

          \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - i \cdot t\right) \cdot b \]

      16. lower-*.f64 63.8

          \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - i \cdot t\right) \cdot b \]

   4. Applied rewrites 63.8%

      \[\leadsto \color{blue}{\left(c \cdot a - i \cdot y\right) \cdot j - \left(c \cdot z - i \cdot t\right) \cdot b} \]

   if 2.2e17 < x

   1. Initial program 73.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in j around 0

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) - b \cdot \left(c \cdot z - i \cdot t\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot t\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot z - a \cdot t\right) \cdot x - \color{blue}{b} \cdot \left(c \cdot z - i \cdot t\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x - \color{blue}{b} \cdot \left(c \cdot z - i \cdot t\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      6. *-commutative N/A

         \[\leadsto \left(z \cdot y - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(z \cdot y - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      8. *-commutative N/A

         \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      10. *-commutative N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot \color{blue}{b} \]

      11. *-commutative N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b \]

      12. lower-*.f64 N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot \color{blue}{b} \]

      13. lift--.f64 N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b \]

      14. lift-*.f64 N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b \]

      15. *-commutative N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot b \]

      16. lower-*.f64 67.3

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot b \]

   4. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot b} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z\right) \cdot b \]
   6. Step-by-step derivation

      1. lift-*.f64 63.8

         \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z\right) \cdot b \]

   7. Applied rewrites 63.8%

      \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z\right) \cdot b \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 59.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-b\right) \cdot \left(c \cdot z\right)\\ t_2 := j \cdot \left(c \cdot a - y \cdot i\right)\\ t_3 := \left(y \cdot x - c \cdot b\right) \cdot z\\ \mathbf{if}\;z \leq -1.55 \cdot 10^{+196}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-39}:\\ \;\;\;\;t\_1 + t\_2\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+15}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b + t\_2\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(-a, x \cdot t, t\_1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (- b) (* c z)))
        (t_2 (* j (- (* c a) (* y i))))
        (t_3 (* (- (* y x) (* c b)) z)))
   (if (<= z -1.55e+196)
     t_3
     (if (<= z -1.25e-39)
       (+ t_1 t_2)
       (if (<= z 2.7e+15)
         (+ (* (* i t) b) t_2)
         (if (<= z 1.25e+64) (fma (- a) (* x t) t_1) t_3))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = -b * (c * z);
	double t_2 = j * ((c * a) - (y * i));
	double t_3 = ((y * x) - (c * b)) * z;
	double tmp;
	if (z <= -1.55e+196) {
		tmp = t_3;
	} else if (z <= -1.25e-39) {
		tmp = t_1 + t_2;
	} else if (z <= 2.7e+15) {
		tmp = ((i * t) * b) + t_2;
	} else if (z <= 1.25e+64) {
		tmp = fma(-a, (x * t), t_1);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(-b) * Float64(c * z))
	t_2 = Float64(j * Float64(Float64(c * a) - Float64(y * i)))
	t_3 = Float64(Float64(Float64(y * x) - Float64(c * b)) * z)
	tmp = 0.0
	if (z <= -1.55e+196)
		tmp = t_3;
	elseif (z <= -1.25e-39)
		tmp = Float64(t_1 + t_2);
	elseif (z <= 2.7e+15)
		tmp = Float64(Float64(Float64(i * t) * b) + t_2);
	elseif (z <= 1.25e+64)
		tmp = fma(Float64(-a), Float64(x * t), t_1);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-b) * N[(c * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(y * x), $MachinePrecision] - N[(c * b), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.55e+196], t$95$3, If[LessEqual[z, -1.25e-39], N[(t$95$1 + t$95$2), $MachinePrecision], If[LessEqual[z, 2.7e+15], N[(N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision] + t$95$2), $MachinePrecision], If[LessEqual[z, 1.25e+64], N[((-a) * N[(x * t), $MachinePrecision] + t$95$1), $MachinePrecision], t$95$3]]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.55000000000000005e196 or 1.25e64 < z

   1. Initial program 63.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

      7. lower-*.f64 67.3

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

   4. Applied rewrites 67.3%

      \[\leadsto \color{blue}{\left(y \cdot x - c \cdot b\right) \cdot z} \]

   if -1.55000000000000005e196 < z < -1.25e-39

   1. Initial program 72.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 74.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in c around inf

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot b\right) \cdot \color{blue}{\left(c \cdot z\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(b\right)\right) \cdot \left(\color{blue}{c} \cdot z\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(-b\right) \cdot \left(\color{blue}{c} \cdot z\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(-b\right) \cdot \color{blue}{\left(c \cdot z\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift-*.f64 50.3

         \[\leadsto \left(-b\right) \cdot \left(c \cdot \color{blue}{z}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   6. Applied rewrites 50.3%

      \[\leadsto \color{blue}{\left(-b\right) \cdot \left(c \cdot z\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   if -1.25e-39 < z < 2.7e15

   1. Initial program 79.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lower-*.f64 59.9

         \[\leadsto \left(i \cdot t\right) \cdot b + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Applied rewrites 59.9%

      \[\leadsto \color{blue}{\left(i \cdot t\right) \cdot b} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   if 2.7e15 < z < 1.25e64

   1. Initial program 77.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)} + a \cdot \left(c \cdot j\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1} \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{t \cdot x}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, \color{blue}{t} \cdot x, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, a \cdot \left(c \cdot j\right) + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(c \cdot j\right) \cdot a + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(c \cdot j, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-1 \cdot b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right)} \]

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(-a, x \cdot t, -1 \cdot \left(b \cdot \left(c \cdot z\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(-1 \cdot b\right) \cdot \left(c \cdot z\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z\right)\right) \]

      4. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(-b\right) \cdot \left(c \cdot z\right)\right) \]

      5. lift-*.f64 38.5

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(-b\right) \cdot \left(c \cdot z\right)\right) \]

   9. Applied rewrites 38.5%

      \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(-b\right) \cdot \left(c \cdot z\right)\right) \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 6: 58.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y \cdot x - c \cdot b\right) \cdot z\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+187}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+15}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b + j \cdot \left(c \cdot a - y \cdot i\right)\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(-a, x \cdot t, \left(-b\right) \cdot \left(c \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (- (* y x) (* c b)) z)))
   (if (<= z -1.8e+187)
     t_1
     (if (<= z 2.7e+15)
       (+ (* (* i t) b) (* j (- (* c a) (* y i))))
       (if (<= z 1.25e+64) (fma (- a) (* x t) (* (- b) (* c z))) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((y * x) - (c * b)) * z;
	double tmp;
	if (z <= -1.8e+187) {
		tmp = t_1;
	} else if (z <= 2.7e+15) {
		tmp = ((i * t) * b) + (j * ((c * a) - (y * i)));
	} else if (z <= 1.25e+64) {
		tmp = fma(-a, (x * t), (-b * (c * z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(y * x) - Float64(c * b)) * z)
	tmp = 0.0
	if (z <= -1.8e+187)
		tmp = t_1;
	elseif (z <= 2.7e+15)
		tmp = Float64(Float64(Float64(i * t) * b) + Float64(j * Float64(Float64(c * a) - Float64(y * i))));
	elseif (z <= 1.25e+64)
		tmp = fma(Float64(-a), Float64(x * t), Float64(Float64(-b) * Float64(c * z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(y * x), $MachinePrecision] - N[(c * b), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.8e+187], t$95$1, If[LessEqual[z, 2.7e+15], N[(N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.25e+64], N[((-a) * N[(x * t), $MachinePrecision] + N[((-b) * N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.80000000000000018e187 or 1.25e64 < z

   1. Initial program 62.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

      7. lower-*.f64 67.2

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

   4. Applied rewrites 67.2%

      \[\leadsto \color{blue}{\left(y \cdot x - c \cdot b\right) \cdot z} \]

   if -1.80000000000000018e187 < z < 2.7e15

   1. Initial program 78.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in i around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lower-*.f64 55.7

         \[\leadsto \left(i \cdot t\right) \cdot b + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Applied rewrites 55.7%

      \[\leadsto \color{blue}{\left(i \cdot t\right) \cdot b} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   if 2.7e15 < z < 1.25e64

   1. Initial program 77.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 78.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)} + a \cdot \left(c \cdot j\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1} \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{t \cdot x}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, \color{blue}{t} \cdot x, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, a \cdot \left(c \cdot j\right) + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(c \cdot j\right) \cdot a + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(c \cdot j, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-1 \cdot b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

   6. Applied rewrites 58.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right)} \]

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(-a, x \cdot t, -1 \cdot \left(b \cdot \left(c \cdot z\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(-1 \cdot b\right) \cdot \left(c \cdot z\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z\right)\right) \]

      4. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(-b\right) \cdot \left(c \cdot z\right)\right) \]

      5. lift-*.f64 38.5

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(-b\right) \cdot \left(c \cdot z\right)\right) \]

   9. Applied rewrites 38.5%

      \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(-b\right) \cdot \left(c \cdot z\right)\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 56.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -3.8 \cdot 10^{+157}:\\ \;\;\;\;\mathsf{fma}\left(-j, y, b \cdot t\right) \cdot i\\ \mathbf{elif}\;i \leq 4.6 \cdot 10^{-219}:\\ \;\;\;\;\left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z\right) \cdot b\\ \mathbf{elif}\;i \leq 2300000000000:\\ \;\;\;\;\left(j \cdot a - b \cdot z\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(b, t, -j \cdot y\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= i -3.8e+157)
   (* (fma (- j) y (* b t)) i)
   (if (<= i 4.6e-219)
     (- (* (- (* z y) (* a t)) x) (* (* c z) b))
     (if (<= i 2300000000000.0)
       (* (- (* j a) (* b z)) c)
       (* (fma b t (- (* j y))) i)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (i <= -3.8e+157) {
		tmp = fma(-j, y, (b * t)) * i;
	} else if (i <= 4.6e-219) {
		tmp = (((z * y) - (a * t)) * x) - ((c * z) * b);
	} else if (i <= 2300000000000.0) {
		tmp = ((j * a) - (b * z)) * c;
	} else {
		tmp = fma(b, t, -(j * y)) * i;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (i <= -3.8e+157)
		tmp = Float64(fma(Float64(-j), y, Float64(b * t)) * i);
	elseif (i <= 4.6e-219)
		tmp = Float64(Float64(Float64(Float64(z * y) - Float64(a * t)) * x) - Float64(Float64(c * z) * b));
	elseif (i <= 2300000000000.0)
		tmp = Float64(Float64(Float64(j * a) - Float64(b * z)) * c);
	else
		tmp = Float64(fma(b, t, Float64(-Float64(j * y))) * i);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[i, -3.8e+157], N[(N[((-j) * y + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[i, 4.6e-219], N[(N[(N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] - N[(N[(c * z), $MachinePrecision] * b), $MachinePrecision]), $MachinePrecision], If[LessEqual[i, 2300000000000.0], N[(N[(N[(j * a), $MachinePrecision] - N[(b * z), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[(b * t + (-N[(j * y), $MachinePrecision])), $MachinePrecision] * i), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if i < -3.8000000000000001e157

   1. Initial program 61.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 63.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right) \cdot \color{blue}{i} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - \left(\mathsf{neg}\left(b\right)\right) \cdot t\right) \cdot i \]

      3. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - \left(-1 \cdot b\right) \cdot t\right) \cdot i \]

      4. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) \cdot i \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) \cdot \color{blue}{i} \]

      6. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - \left(-1 \cdot b\right) \cdot t\right) \cdot i \]

      7. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - \left(\mathsf{neg}\left(b\right)\right) \cdot t\right) \cdot i \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right) \cdot i \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot j\right) \cdot y + b \cdot t\right) \cdot i \]

      10. mul-1-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(j\right)\right) \cdot y + b \cdot t\right) \cdot i \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(j\right), y, b \cdot t\right) \cdot i \]

      12. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-j, y, b \cdot t\right) \cdot i \]

      13. lower-*.f64 69.8

          \[\leadsto \mathsf{fma}\left(-j, y, b \cdot t\right) \cdot i \]

   6. Applied rewrites 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, y, b \cdot t\right) \cdot i} \]

   if -3.8000000000000001e157 < i < 4.59999999999999977e-219

   1. Initial program 77.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in j around 0

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) - b \cdot \left(c \cdot z - i \cdot t\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot t\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot z - a \cdot t\right) \cdot x - \color{blue}{b} \cdot \left(c \cdot z - i \cdot t\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x - \color{blue}{b} \cdot \left(c \cdot z - i \cdot t\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      6. *-commutative N/A

         \[\leadsto \left(z \cdot y - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(z \cdot y - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      8. *-commutative N/A

         \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      10. *-commutative N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot \color{blue}{b} \]

      11. *-commutative N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b \]

      12. lower-*.f64 N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot \color{blue}{b} \]

      13. lift--.f64 N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b \]

      14. lift-*.f64 N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b \]

      15. *-commutative N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot b \]

      16. lower-*.f64 61.9

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot b \]

   4. Applied rewrites 61.9%

      \[\leadsto \color{blue}{\left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot b} \]

   5. Taylor expanded in z around inf

      \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z\right) \cdot b \]
   6. Step-by-step derivation

      1. lift-*.f64 55.4

         \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z\right) \cdot b \]

   7. Applied rewrites 55.4%

      \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z\right) \cdot b \]

   if 4.59999999999999977e-219 < i < 2.3e12

   1. Initial program 79.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot j - b \cdot z\right) \cdot \color{blue}{c} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot j - b \cdot z\right) \cdot \color{blue}{c} \]

      3. lower--.f64 N/A

         \[\leadsto \left(a \cdot j - b \cdot z\right) \cdot c \]

      4. *-commutative N/A

         \[\leadsto \left(j \cdot a - b \cdot z\right) \cdot c \]

      5. lower-*.f64 N/A

         \[\leadsto \left(j \cdot a - b \cdot z\right) \cdot c \]

      6. lower-*.f64 44.3

         \[\leadsto \left(j \cdot a - b \cdot z\right) \cdot c \]

   4. Applied rewrites 44.3%

      \[\leadsto \color{blue}{\left(j \cdot a - b \cdot z\right) \cdot c} \]

   if 2.3e12 < i

   1. Initial program 67.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in j around 0

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right) - b \cdot \left(c \cdot z - i \cdot t\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto x \cdot \left(y \cdot z - a \cdot t\right) - \color{blue}{b \cdot \left(c \cdot z - i \cdot t\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot z - a \cdot t\right) \cdot x - \color{blue}{b} \cdot \left(c \cdot z - i \cdot t\right) \]

      3. *-commutative N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x - \color{blue}{b} \cdot \left(c \cdot z - i \cdot t\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      6. *-commutative N/A

         \[\leadsto \left(z \cdot y - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \left(z \cdot y - t \cdot a\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      8. *-commutative N/A

         \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - b \cdot \left(c \cdot z - i \cdot t\right) \]

      10. *-commutative N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot \color{blue}{b} \]

      11. *-commutative N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b \]

      12. lower-*.f64 N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot \color{blue}{b} \]

      13. lift--.f64 N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b \]

      14. lift-*.f64 N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - t \cdot i\right) \cdot b \]

      15. *-commutative N/A

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot b \]

      16. lower-*.f64 53.7

          \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot b \]

   4. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\left(z \cdot y - a \cdot t\right) \cdot x - \left(c \cdot z - i \cdot t\right) \cdot b} \]

   5. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto i \cdot \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) \cdot \color{blue}{i} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) \cdot \color{blue}{i} \]

      4. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - \left(-1 \cdot b\right) \cdot t\right) \cdot i \]

      5. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - \left(\mathsf{neg}\left(b\right)\right) \cdot t\right) \cdot i \]

      6. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right) \cdot i \]

      7. +-commutative N/A

         \[\leadsto \left(b \cdot t + -1 \cdot \left(j \cdot y\right)\right) \cdot i \]

      8. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(b, t, -1 \cdot \left(j \cdot y\right)\right) \cdot i \]

      9. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(b, t, \mathsf{neg}\left(j \cdot y\right)\right) \cdot i \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(b, t, -j \cdot y\right) \cdot i \]

      11. lower-*.f64 60.4

          \[\leadsto \mathsf{fma}\left(b, t, -j \cdot y\right) \cdot i \]

   7. Applied rewrites 60.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(b, t, -j \cdot y\right) \cdot i} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 8: 52.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.3 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(-j, y, b \cdot t\right) \cdot i\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-97}:\\ \;\;\;\;\mathsf{fma}\left(-a, x \cdot t, \left(-b\right) \cdot \left(c \cdot z\right)\right)\\ \mathbf{elif}\;y \leq 1.32 \cdot 10^{-23}:\\ \;\;\;\;\left(j \cdot a - b \cdot z\right) \cdot c\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -6.3e+153)
   (* (fma (- j) y (* b t)) i)
   (if (<= y -4.8e-97)
     (fma (- a) (* x t) (* (- b) (* c z)))
     (if (<= y 1.32e-23)
       (* (- (* j a) (* b z)) c)
       (* (fma (- i) j (* z x)) y)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -6.3e+153) {
		tmp = fma(-j, y, (b * t)) * i;
	} else if (y <= -4.8e-97) {
		tmp = fma(-a, (x * t), (-b * (c * z)));
	} else if (y <= 1.32e-23) {
		tmp = ((j * a) - (b * z)) * c;
	} else {
		tmp = fma(-i, j, (z * x)) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -6.3e+153)
		tmp = Float64(fma(Float64(-j), y, Float64(b * t)) * i);
	elseif (y <= -4.8e-97)
		tmp = fma(Float64(-a), Float64(x * t), Float64(Float64(-b) * Float64(c * z)));
	elseif (y <= 1.32e-23)
		tmp = Float64(Float64(Float64(j * a) - Float64(b * z)) * c);
	else
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -6.3e+153], N[(N[((-j) * y + N[(b * t), $MachinePrecision]), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[y, -4.8e-97], N[((-a) * N[(x * t), $MachinePrecision] + N[((-b) * N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.32e-23], N[(N[(N[(j * a), $MachinePrecision] - N[(b * z), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -6.3000000000000001e153

   1. Initial program 59.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in i around inf

      \[\leadsto \color{blue}{i \cdot \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right) \cdot \color{blue}{i} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - \left(\mathsf{neg}\left(b\right)\right) \cdot t\right) \cdot i \]

      3. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - \left(-1 \cdot b\right) \cdot t\right) \cdot i \]

      4. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) \cdot i \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - -1 \cdot \left(b \cdot t\right)\right) \cdot \color{blue}{i} \]

      6. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - \left(-1 \cdot b\right) \cdot t\right) \cdot i \]

      7. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) - \left(\mathsf{neg}\left(b\right)\right) \cdot t\right) \cdot i \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(-1 \cdot \left(j \cdot y\right) + b \cdot t\right) \cdot i \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot j\right) \cdot y + b \cdot t\right) \cdot i \]

      10. mul-1-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(j\right)\right) \cdot y + b \cdot t\right) \cdot i \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(j\right), y, b \cdot t\right) \cdot i \]

      12. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-j, y, b \cdot t\right) \cdot i \]

      13. lower-*.f64 48.6

          \[\leadsto \mathsf{fma}\left(-j, y, b \cdot t\right) \cdot i \]

   6. Applied rewrites 48.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-j, y, b \cdot t\right) \cdot i} \]

   if -6.3000000000000001e153 < y < -4.8e-97

   1. Initial program 75.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)} + a \cdot \left(c \cdot j\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1} \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{t \cdot x}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, \color{blue}{t} \cdot x, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, a \cdot \left(c \cdot j\right) + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(c \cdot j\right) \cdot a + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(c \cdot j, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-1 \cdot b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

   6. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right)} \]

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{fma}\left(-a, x \cdot t, -1 \cdot \left(b \cdot \left(c \cdot z\right)\right)\right) \]
   8. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(-1 \cdot b\right) \cdot \left(c \cdot z\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z\right)\right) \]

      3. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z\right)\right) \]

      4. lift-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(-b\right) \cdot \left(c \cdot z\right)\right) \]

      5. lift-*.f64 37.2

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(-b\right) \cdot \left(c \cdot z\right)\right) \]

   9. Applied rewrites 37.2%

      \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(-b\right) \cdot \left(c \cdot z\right)\right) \]

   if -4.8e-97 < y < 1.31999999999999994e-23

   1. Initial program 81.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in c around inf

      \[\leadsto \color{blue}{c \cdot \left(a \cdot j - b \cdot z\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot j - b \cdot z\right) \cdot \color{blue}{c} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot j - b \cdot z\right) \cdot \color{blue}{c} \]

      3. lower--.f64 N/A

         \[\leadsto \left(a \cdot j - b \cdot z\right) \cdot c \]

      4. *-commutative N/A

         \[\leadsto \left(j \cdot a - b \cdot z\right) \cdot c \]

      5. lower-*.f64 N/A

         \[\leadsto \left(j \cdot a - b \cdot z\right) \cdot c \]

      6. lower-*.f64 46.7

         \[\leadsto \left(j \cdot a - b \cdot z\right) \cdot c \]

   4. Applied rewrites 46.7%

      \[\leadsto \color{blue}{\left(j \cdot a - b \cdot z\right) \cdot c} \]

   if 1.31999999999999994e-23 < y

   1. Initial program 66.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in y around inf

      \[\leadsto \color{blue}{y \cdot \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot \color{blue}{y} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(i \cdot j\right) + x \cdot z\right) \cdot \color{blue}{y} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot i\right) \cdot j + x \cdot z\right) \cdot y \]

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(i\right)\right) \cdot j + x \cdot z\right) \cdot y \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(i\right), j, x \cdot z\right) \cdot y \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-i, j, x \cdot z\right) \cdot y \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y \]

      8. lower-*.f64 58.2

         \[\leadsto \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y \]

   4. Applied rewrites 58.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 9: 52.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{if}\;j \leq -1.52 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq -2.1 \cdot 10^{-180}:\\ \;\;\;\;\left(y \cdot x - c \cdot b\right) \cdot z\\ \mathbf{elif}\;j \leq 3.3 \cdot 10^{-58}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (- (* c a) (* i y)) j)))
   (if (<= j -1.52e+33)
     t_1
     (if (<= j -2.1e-180)
       (* (- (* y x) (* c b)) z)
       (if (<= j 3.3e-58) (* (fma (- a) x (* i b)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((c * a) - (i * y)) * j;
	double tmp;
	if (j <= -1.52e+33) {
		tmp = t_1;
	} else if (j <= -2.1e-180) {
		tmp = ((y * x) - (c * b)) * z;
	} else if (j <= 3.3e-58) {
		tmp = fma(-a, x, (i * b)) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(c * a) - Float64(i * y)) * j)
	tmp = 0.0
	if (j <= -1.52e+33)
		tmp = t_1;
	elseif (j <= -2.1e-180)
		tmp = Float64(Float64(Float64(y * x) - Float64(c * b)) * z);
	elseif (j <= 3.3e-58)
		tmp = Float64(fma(Float64(-a), x, Float64(i * b)) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]}, If[LessEqual[j, -1.52e+33], t$95$1, If[LessEqual[j, -2.1e-180], N[(N[(N[(y * x), $MachinePrecision] - N[(c * b), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[j, 3.3e-58], N[(N[((-a) * x + N[(i * b), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if j < -1.5200000000000001e33 or 3.30000000000000026e-58 < j

   1. Initial program 73.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot \color{blue}{j} \]

      2. *-commutative N/A

         \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j \]

      3. *-commutative N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j \]

      4. lower-*.f64 N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot \color{blue}{j} \]

      5. lift--.f64 N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j \]

      6. lift-*.f64 N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j \]

      7. *-commutative N/A

         \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j \]

      8. lower-*.f64 58.5

         \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j \]

   4. Applied rewrites 58.5%

      \[\leadsto \color{blue}{\left(c \cdot a - i \cdot y\right) \cdot j} \]

   if -1.5200000000000001e33 < j < -2.0999999999999999e-180

   1. Initial program 73.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

      7. lower-*.f64 43.0

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

   4. Applied rewrites 43.0%

      \[\leadsto \color{blue}{\left(y \cdot x - c \cdot b\right) \cdot z} \]

   if -2.0999999999999999e-180 < j < 3.30000000000000026e-58

   1. Initial program 73.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 74.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + b \cdot i\right)} \]
   5. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + b \cdot i\right) \cdot \color{blue}{t} \]

      2. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) - \left(\mathsf{neg}\left(b\right)\right) \cdot i\right) \cdot t \]

      3. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) - \left(-1 \cdot b\right) \cdot i\right) \cdot t \]

      4. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot t \]

      5. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) - -1 \cdot \left(b \cdot i\right)\right) \cdot \color{blue}{t} \]

      6. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) - \left(-1 \cdot b\right) \cdot i\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) - \left(\mathsf{neg}\left(b\right)\right) \cdot i\right) \cdot t \]

      8. fp-cancel-sign-sub-inv N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + b \cdot i\right) \cdot t \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot a\right) \cdot x + b \cdot i\right) \cdot t \]

      10. mul-1-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(a\right)\right) \cdot x + b \cdot i\right) \cdot t \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), x, b \cdot i\right) \cdot t \]

      12. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x, b \cdot i\right) \cdot t \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t \]

      14. lower-*.f64 46.9

          \[\leadsto \mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t \]

   6. Applied rewrites 46.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, i \cdot b\right) \cdot t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 48.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(i \cdot t - c \cdot z\right) \cdot b\\ \mathbf{if}\;b \leq -1.8 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 4.7 \cdot 10^{+45}:\\ \;\;\;\;\left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (- (* i t) (* c z)) b)))
   (if (<= b -1.8e+22) t_1 (if (<= b 4.7e+45) (* (- (* c a) (* i y)) j) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((i * t) - (c * z)) * b;
	double tmp;
	if (b <= -1.8e+22) {
		tmp = t_1;
	} else if (b <= 4.7e+45) {
		tmp = ((c * a) - (i * y)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((i * t) - (c * z)) * b
    if (b <= (-1.8d+22)) then
        tmp = t_1
    else if (b <= 4.7d+45) then
        tmp = ((c * a) - (i * y)) * j
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((i * t) - (c * z)) * b;
	double tmp;
	if (b <= -1.8e+22) {
		tmp = t_1;
	} else if (b <= 4.7e+45) {
		tmp = ((c * a) - (i * y)) * j;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((i * t) - (c * z)) * b
	tmp = 0
	if b <= -1.8e+22:
		tmp = t_1
	elif b <= 4.7e+45:
		tmp = ((c * a) - (i * y)) * j
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(i * t) - Float64(c * z)) * b)
	tmp = 0.0
	if (b <= -1.8e+22)
		tmp = t_1;
	elseif (b <= 4.7e+45)
		tmp = Float64(Float64(Float64(c * a) - Float64(i * y)) * j);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((i * t) - (c * z)) * b;
	tmp = 0.0;
	if (b <= -1.8e+22)
		tmp = t_1;
	elseif (b <= 4.7e+45)
		tmp = ((c * a) - (i * y)) * j;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(i * t), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[b, -1.8e+22], t$95$1, If[LessEqual[b, 4.7e+45], N[(N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -1.8e22 or 4.70000000000000002e45 < b

   1. Initial program 73.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(i \cdot t - c \cdot z\right) \cdot \color{blue}{b} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t - c \cdot z\right) \cdot \color{blue}{b} \]

      3. lower--.f64 N/A

         \[\leadsto \left(i \cdot t - c \cdot z\right) \cdot b \]

      4. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t - c \cdot z\right) \cdot b \]

      5. lift-*.f64 61.8

         \[\leadsto \left(i \cdot t - c \cdot z\right) \cdot b \]

   4. Applied rewrites 61.8%

      \[\leadsto \color{blue}{\left(i \cdot t - c \cdot z\right) \cdot b} \]

   if -1.8e22 < b < 4.70000000000000002e45

   1. Initial program 73.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot \color{blue}{j} \]

      2. *-commutative N/A

         \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j \]

      3. *-commutative N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j \]

      4. lower-*.f64 N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot \color{blue}{j} \]

      5. lift--.f64 N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j \]

      6. lift-*.f64 N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j \]

      7. *-commutative N/A

         \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j \]

      8. lower-*.f64 45.2

         \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j \]

   4. Applied rewrites 45.2%

      \[\leadsto \color{blue}{\left(c \cdot a - i \cdot y\right) \cdot j} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 41.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7 \cdot 10^{+76}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;b \leq 4.3 \cdot 10^{+72}:\\ \;\;\;\;\left(c \cdot a - i \cdot y\right) \cdot j\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(c \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -7e+76)
   (* (* i b) t)
   (if (<= b 4.3e+72) (* (- (* c a) (* i y)) j) (* (- b) (* c z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -7e+76) {
		tmp = (i * b) * t;
	} else if (b <= 4.3e+72) {
		tmp = ((c * a) - (i * y)) * j;
	} else {
		tmp = -b * (c * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (b <= (-7d+76)) then
        tmp = (i * b) * t
    else if (b <= 4.3d+72) then
        tmp = ((c * a) - (i * y)) * j
    else
        tmp = -b * (c * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -7e+76) {
		tmp = (i * b) * t;
	} else if (b <= 4.3e+72) {
		tmp = ((c * a) - (i * y)) * j;
	} else {
		tmp = -b * (c * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -7e+76:
		tmp = (i * b) * t
	elif b <= 4.3e+72:
		tmp = ((c * a) - (i * y)) * j
	else:
		tmp = -b * (c * z)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -7e+76)
		tmp = Float64(Float64(i * b) * t);
	elseif (b <= 4.3e+72)
		tmp = Float64(Float64(Float64(c * a) - Float64(i * y)) * j);
	else
		tmp = Float64(Float64(-b) * Float64(c * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -7e+76)
		tmp = (i * b) * t;
	elseif (b <= 4.3e+72)
		tmp = ((c * a) - (i * y)) * j;
	else
		tmp = -b * (c * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -7e+76], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[b, 4.3e+72], N[(N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision], N[((-b) * N[(c * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.00000000000000001e76

   1. Initial program 74.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 77.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)} + a \cdot \left(c \cdot j\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1} \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{t \cdot x}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, \color{blue}{t} \cdot x, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, a \cdot \left(c \cdot j\right) + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(c \cdot j\right) \cdot a + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(c \cdot j, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-1 \cdot b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

   6. Applied rewrites 74.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right)} \]

   7. Taylor expanded in i around inf

      \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      3. lift-*.f64 36.5

         \[\leadsto \left(i \cdot t\right) \cdot b \]

   9. Applied rewrites 36.5%

      \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(i \cdot t\right) \cdot b \]

       2. lift-*.f64 N/A

          \[\leadsto \left(i \cdot t\right) \cdot b \]

       3. *-commutative N/A

          \[\leadsto b \cdot \left(i \cdot \color{blue}{t}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(b \cdot i\right) \cdot t \]

       5. lower-*.f64 N/A

          \[\leadsto \left(b \cdot i\right) \cdot t \]

       6. *-commutative N/A

          \[\leadsto \left(i \cdot b\right) \cdot t \]

       7. lower-*.f64 37.3

          \[\leadsto \left(i \cdot b\right) \cdot t \]

   11. Applied rewrites 37.3%

       \[\leadsto \left(i \cdot b\right) \cdot t \]

   if -7.00000000000000001e76 < b < 4.3000000000000001e72

   1. Initial program 73.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in j around inf

      \[\leadsto \color{blue}{j \cdot \left(a \cdot c - i \cdot y\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot c - i \cdot y\right) \cdot \color{blue}{j} \]

      2. *-commutative N/A

         \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j \]

      3. *-commutative N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j \]

      4. lower-*.f64 N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot \color{blue}{j} \]

      5. lift--.f64 N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j \]

      6. lift-*.f64 N/A

         \[\leadsto \left(c \cdot a - y \cdot i\right) \cdot j \]

      7. *-commutative N/A

         \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j \]

      8. lower-*.f64 44.0

         \[\leadsto \left(c \cdot a - i \cdot y\right) \cdot j \]

   4. Applied rewrites 44.0%

      \[\leadsto \color{blue}{\left(c \cdot a - i \cdot y\right) \cdot j} \]

   if 4.3000000000000001e72 < b

   1. Initial program 71.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

      7. lower-*.f64 46.5

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

   4. Applied rewrites 46.5%

      \[\leadsto \color{blue}{\left(y \cdot x - c \cdot b\right) \cdot z} \]

   5. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot b\right) \cdot \left(c \cdot \color{blue}{z}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z\right) \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(-b\right) \cdot \left(c \cdot z\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(-b\right) \cdot \left(c \cdot \color{blue}{z}\right) \]

      5. lift-*.f64 36.8

         \[\leadsto \left(-b\right) \cdot \left(c \cdot z\right) \]

   7. Applied rewrites 36.8%

      \[\leadsto \left(-b\right) \cdot \color{blue}{\left(c \cdot z\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 29.8% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -1350000000000:\\ \;\;\;\;\left(-b\right) \cdot \left(c \cdot z\right)\\ \mathbf{elif}\;c \leq 1.7 \cdot 10^{-241}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{+73}:\\ \;\;\;\;\left(\left(-t\right) \cdot x\right) \cdot a\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot c\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -1350000000000.0)
   (* (- b) (* c z))
   (if (<= c 1.7e-241)
     (* (* i t) b)
     (if (<= c 2.7e+73) (* (* (- t) x) a) (* (* (- b) c) z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -1350000000000.0) {
		tmp = -b * (c * z);
	} else if (c <= 1.7e-241) {
		tmp = (i * t) * b;
	} else if (c <= 2.7e+73) {
		tmp = (-t * x) * a;
	} else {
		tmp = (-b * c) * z;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (c <= (-1350000000000.0d0)) then
        tmp = -b * (c * z)
    else if (c <= 1.7d-241) then
        tmp = (i * t) * b
    else if (c <= 2.7d+73) then
        tmp = (-t * x) * a
    else
        tmp = (-b * c) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (c <= -1350000000000.0) {
		tmp = -b * (c * z);
	} else if (c <= 1.7e-241) {
		tmp = (i * t) * b;
	} else if (c <= 2.7e+73) {
		tmp = (-t * x) * a;
	} else {
		tmp = (-b * c) * z;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if c <= -1350000000000.0:
		tmp = -b * (c * z)
	elif c <= 1.7e-241:
		tmp = (i * t) * b
	elif c <= 2.7e+73:
		tmp = (-t * x) * a
	else:
		tmp = (-b * c) * z
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -1350000000000.0)
		tmp = Float64(Float64(-b) * Float64(c * z));
	elseif (c <= 1.7e-241)
		tmp = Float64(Float64(i * t) * b);
	elseif (c <= 2.7e+73)
		tmp = Float64(Float64(Float64(-t) * x) * a);
	else
		tmp = Float64(Float64(Float64(-b) * c) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (c <= -1350000000000.0)
		tmp = -b * (c * z);
	elseif (c <= 1.7e-241)
		tmp = (i * t) * b;
	elseif (c <= 2.7e+73)
		tmp = (-t * x) * a;
	else
		tmp = (-b * c) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -1350000000000.0], N[((-b) * N[(c * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 1.7e-241], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[c, 2.7e+73], N[(N[((-t) * x), $MachinePrecision] * a), $MachinePrecision], N[(N[((-b) * c), $MachinePrecision] * z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -1.35e12

   1. Initial program 68.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

      7. lower-*.f64 46.4

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

   4. Applied rewrites 46.4%

      \[\leadsto \color{blue}{\left(y \cdot x - c \cdot b\right) \cdot z} \]

   5. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot b\right) \cdot \left(c \cdot \color{blue}{z}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z\right) \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(-b\right) \cdot \left(c \cdot z\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(-b\right) \cdot \left(c \cdot \color{blue}{z}\right) \]

      5. lift-*.f64 35.5

         \[\leadsto \left(-b\right) \cdot \left(c \cdot z\right) \]

   7. Applied rewrites 35.5%

      \[\leadsto \left(-b\right) \cdot \color{blue}{\left(c \cdot z\right)} \]

   if -1.35e12 < c < 1.6999999999999999e-241

   1. Initial program 80.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 81.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)} + a \cdot \left(c \cdot j\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1} \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{t \cdot x}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, \color{blue}{t} \cdot x, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, a \cdot \left(c \cdot j\right) + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(c \cdot j\right) \cdot a + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(c \cdot j, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-1 \cdot b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

   6. Applied rewrites 55.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right)} \]

   7. Taylor expanded in i around inf

      \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      3. lift-*.f64 25.5

         \[\leadsto \left(i \cdot t\right) \cdot b \]

   9. Applied rewrites 25.5%

      \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]

   if 1.6999999999999999e-241 < c < 2.6999999999999999e73

   1. Initial program 78.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right) \cdot \color{blue}{a} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot t\right) \cdot x + c \cdot j\right) \cdot a \]

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(t\right)\right) \cdot x + c \cdot j\right) \cdot a \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(t\right), x, c \cdot j\right) \cdot a \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-t, x, c \cdot j\right) \cdot a \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-t, x, j \cdot c\right) \cdot a \]

      8. lower-*.f64 34.7

         \[\leadsto \mathsf{fma}\left(-t, x, j \cdot c\right) \cdot a \]

   4. Applied rewrites 34.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-t, x, j \cdot c\right) \cdot a} \]

   5. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot \left(t \cdot x\right)\right) \cdot a \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(t \cdot x\right)\right) \cdot a \]

      2. distribute-lft-neg-out N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(t\right)\right) \cdot x\right) \cdot a \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(\left(-t\right) \cdot x\right) \cdot a \]

      4. lower-*.f64 24.6

         \[\leadsto \left(\left(-t\right) \cdot x\right) \cdot a \]

   7. Applied rewrites 24.6%

      \[\leadsto \left(\left(-t\right) \cdot x\right) \cdot a \]

   if 2.6999999999999999e73 < c

   1. Initial program 62.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

      7. lower-*.f64 46.3

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

   4. Applied rewrites 46.3%

      \[\leadsto \color{blue}{\left(y \cdot x - c \cdot b\right) \cdot z} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(b \cdot c\right)\right) \cdot z \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot b\right) \cdot c\right) \cdot z \]

      2. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(b\right)\right) \cdot c\right) \cdot z \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(\left(-b\right) \cdot c\right) \cdot z \]

      4. lower-*.f64 36.7

         \[\leadsto \left(\left(-b\right) \cdot c\right) \cdot z \]

   7. Applied rewrites 36.7%

      \[\leadsto \left(\left(-b\right) \cdot c\right) \cdot z \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 13: 29.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -95000000000:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{+27}:\\ \;\;\;\;\left(\left(-a\right) \cdot t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(c \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -95000000000.0)
   (* (* i b) t)
   (if (<= b 7.5e+27) (* (* (- a) t) x) (* (- b) (* c z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -95000000000.0) {
		tmp = (i * b) * t;
	} else if (b <= 7.5e+27) {
		tmp = (-a * t) * x;
	} else {
		tmp = -b * (c * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (b <= (-95000000000.0d0)) then
        tmp = (i * b) * t
    else if (b <= 7.5d+27) then
        tmp = (-a * t) * x
    else
        tmp = -b * (c * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -95000000000.0) {
		tmp = (i * b) * t;
	} else if (b <= 7.5e+27) {
		tmp = (-a * t) * x;
	} else {
		tmp = -b * (c * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -95000000000.0:
		tmp = (i * b) * t
	elif b <= 7.5e+27:
		tmp = (-a * t) * x
	else:
		tmp = -b * (c * z)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -95000000000.0)
		tmp = Float64(Float64(i * b) * t);
	elseif (b <= 7.5e+27)
		tmp = Float64(Float64(Float64(-a) * t) * x);
	else
		tmp = Float64(Float64(-b) * Float64(c * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -95000000000.0)
		tmp = (i * b) * t;
	elseif (b <= 7.5e+27)
		tmp = (-a * t) * x;
	else
		tmp = -b * (c * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -95000000000.0], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[b, 7.5e+27], N[(N[((-a) * t), $MachinePrecision] * x), $MachinePrecision], N[((-b) * N[(c * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.5e10

   1. Initial program 74.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)} + a \cdot \left(c \cdot j\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1} \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{t \cdot x}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, \color{blue}{t} \cdot x, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, a \cdot \left(c \cdot j\right) + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(c \cdot j\right) \cdot a + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(c \cdot j, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-1 \cdot b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

   6. Applied rewrites 71.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right)} \]

   7. Taylor expanded in i around inf

      \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      3. lift-*.f64 33.0

         \[\leadsto \left(i \cdot t\right) \cdot b \]

   9. Applied rewrites 33.0%

      \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(i \cdot t\right) \cdot b \]

       2. lift-*.f64 N/A

          \[\leadsto \left(i \cdot t\right) \cdot b \]

       3. *-commutative N/A

          \[\leadsto b \cdot \left(i \cdot \color{blue}{t}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(b \cdot i\right) \cdot t \]

       5. lower-*.f64 N/A

          \[\leadsto \left(b \cdot i\right) \cdot t \]

       6. *-commutative N/A

          \[\leadsto \left(i \cdot b\right) \cdot t \]

       7. lower-*.f64 33.5

          \[\leadsto \left(i \cdot b\right) \cdot t \]

   11. Applied rewrites 33.5%

       \[\leadsto \left(i \cdot b\right) \cdot t \]

   if -9.5e10 < b < 7.5000000000000002e27

   1. Initial program 73.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot z - a \cdot t\right) \cdot \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x \]

      3. lower-*.f64 N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot \color{blue}{x} \]

      4. lift--.f64 N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(z \cdot y - t \cdot a\right) \cdot x \]

      6. lower-*.f64 N/A

         \[\leadsto \left(z \cdot y - t \cdot a\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x \]

      8. lower-*.f64 45.8

         \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x \]

   4. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\left(z \cdot y - a \cdot t\right) \cdot x} \]

   5. Taylor expanded in y around 0

      \[\leadsto \left(-1 \cdot \left(a \cdot t\right)\right) \cdot x \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot a\right) \cdot t\right) \cdot x \]

      2. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(a\right)\right) \cdot t\right) \cdot x \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(\left(-a\right) \cdot t\right) \cdot x \]

      4. lower-*.f64 24.9

         \[\leadsto \left(\left(-a\right) \cdot t\right) \cdot x \]

   7. Applied rewrites 24.9%

      \[\leadsto \left(\left(-a\right) \cdot t\right) \cdot x \]

   if 7.5000000000000002e27 < b

   1. Initial program 71.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

      7. lower-*.f64 45.4

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

   4. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\left(y \cdot x - c \cdot b\right) \cdot z} \]

   5. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot b\right) \cdot \left(c \cdot \color{blue}{z}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z\right) \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(-b\right) \cdot \left(c \cdot z\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(-b\right) \cdot \left(c \cdot \color{blue}{z}\right) \]

      5. lift-*.f64 34.8

         \[\leadsto \left(-b\right) \cdot \left(c \cdot z\right) \]

   7. Applied rewrites 34.8%

      \[\leadsto \left(-b\right) \cdot \color{blue}{\left(c \cdot z\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 29.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -95000000000:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{elif}\;b \leq 7 \cdot 10^{+27}:\\ \;\;\;\;\left(\left(-a\right) \cdot x\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(-b\right) \cdot \left(c \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= b -95000000000.0)
   (* (* i b) t)
   (if (<= b 7e+27) (* (* (- a) x) t) (* (- b) (* c z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -95000000000.0) {
		tmp = (i * b) * t;
	} else if (b <= 7e+27) {
		tmp = (-a * x) * t;
	} else {
		tmp = -b * (c * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (b <= (-95000000000.0d0)) then
        tmp = (i * b) * t
    else if (b <= 7d+27) then
        tmp = (-a * x) * t
    else
        tmp = -b * (c * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (b <= -95000000000.0) {
		tmp = (i * b) * t;
	} else if (b <= 7e+27) {
		tmp = (-a * x) * t;
	} else {
		tmp = -b * (c * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if b <= -95000000000.0:
		tmp = (i * b) * t
	elif b <= 7e+27:
		tmp = (-a * x) * t
	else:
		tmp = -b * (c * z)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (b <= -95000000000.0)
		tmp = Float64(Float64(i * b) * t);
	elseif (b <= 7e+27)
		tmp = Float64(Float64(Float64(-a) * x) * t);
	else
		tmp = Float64(Float64(-b) * Float64(c * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (b <= -95000000000.0)
		tmp = (i * b) * t;
	elseif (b <= 7e+27)
		tmp = (-a * x) * t;
	else
		tmp = -b * (c * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[b, -95000000000.0], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[b, 7e+27], N[(N[((-a) * x), $MachinePrecision] * t), $MachinePrecision], N[((-b) * N[(c * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -9.5e10

   1. Initial program 74.5%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)} + a \cdot \left(c \cdot j\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1} \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{t \cdot x}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, \color{blue}{t} \cdot x, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, a \cdot \left(c \cdot j\right) + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(c \cdot j\right) \cdot a + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(c \cdot j, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-1 \cdot b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

   6. Applied rewrites 71.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right)} \]

   7. Taylor expanded in i around inf

      \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      3. lift-*.f64 33.0

         \[\leadsto \left(i \cdot t\right) \cdot b \]

   9. Applied rewrites 33.0%

      \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(i \cdot t\right) \cdot b \]

       2. lift-*.f64 N/A

          \[\leadsto \left(i \cdot t\right) \cdot b \]

       3. *-commutative N/A

          \[\leadsto b \cdot \left(i \cdot \color{blue}{t}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(b \cdot i\right) \cdot t \]

       5. lower-*.f64 N/A

          \[\leadsto \left(b \cdot i\right) \cdot t \]

       6. *-commutative N/A

          \[\leadsto \left(i \cdot b\right) \cdot t \]

       7. lower-*.f64 33.5

          \[\leadsto \left(i \cdot b\right) \cdot t \]

   11. Applied rewrites 33.5%

       \[\leadsto \left(i \cdot b\right) \cdot t \]

   if -9.5e10 < b < 7.0000000000000004e27

   1. Initial program 73.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(y \cdot z - a \cdot t\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot z - a \cdot t\right) \cdot \color{blue}{x} \]

      2. *-commutative N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x \]

      3. lower-*.f64 N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot \color{blue}{x} \]

      4. lift--.f64 N/A

         \[\leadsto \left(y \cdot z - t \cdot a\right) \cdot x \]

      5. *-commutative N/A

         \[\leadsto \left(z \cdot y - t \cdot a\right) \cdot x \]

      6. lower-*.f64 N/A

         \[\leadsto \left(z \cdot y - t \cdot a\right) \cdot x \]

      7. *-commutative N/A

         \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x \]

      8. lower-*.f64 45.8

         \[\leadsto \left(z \cdot y - a \cdot t\right) \cdot x \]

   4. Applied rewrites 45.8%

      \[\leadsto \color{blue}{\left(z \cdot y - a \cdot t\right) \cdot x} \]

   5. Taylor expanded in y around 0

      \[\leadsto -1 \cdot \color{blue}{\left(a \cdot \left(t \cdot x\right)\right)} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(a \cdot \left(t \cdot x\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg}\left(a \cdot \left(x \cdot t\right)\right) \]

      3. distribute-lft-neg-out N/A

         \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(x \cdot \color{blue}{t}\right) \]

      4. lift-neg.f64 N/A

         \[\leadsto \left(-a\right) \cdot \left(x \cdot t\right) \]

      5. associate-*r* N/A

         \[\leadsto \left(\left(-a\right) \cdot x\right) \cdot t \]

      6. lift-neg.f64 N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(a\right)\right) \cdot x\right) \cdot t \]

      7. mul-1-neg N/A

         \[\leadsto \left(\left(-1 \cdot a\right) \cdot x\right) \cdot t \]

      8. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right)\right) \cdot t \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right)\right) \cdot t \]

      10. associate-*r* N/A

          \[\leadsto \left(\left(-1 \cdot a\right) \cdot x\right) \cdot t \]

      11. mul-1-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(a\right)\right) \cdot x\right) \cdot t \]

      12. lift-neg.f64 N/A

          \[\leadsto \left(\left(-a\right) \cdot x\right) \cdot t \]

      13. lower-*.f64 25.1

          \[\leadsto \left(\left(-a\right) \cdot x\right) \cdot t \]

   7. Applied rewrites 25.1%

      \[\leadsto \left(\left(-a\right) \cdot x\right) \cdot \color{blue}{t} \]

   if 7.0000000000000004e27 < b

   1. Initial program 71.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

      7. lower-*.f64 45.4

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

   4. Applied rewrites 45.4%

      \[\leadsto \color{blue}{\left(y \cdot x - c \cdot b\right) \cdot z} \]

   5. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot b\right) \cdot \left(c \cdot \color{blue}{z}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z\right) \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(-b\right) \cdot \left(c \cdot z\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(-b\right) \cdot \left(c \cdot \color{blue}{z}\right) \]

      5. lift-*.f64 34.8

         \[\leadsto \left(-b\right) \cdot \left(c \cdot z\right) \]

   7. Applied rewrites 34.8%

      \[\leadsto \left(-b\right) \cdot \color{blue}{\left(c \cdot z\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 29.5% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-b\right) \cdot \left(c \cdot z\right)\\ \mathbf{if}\;c \leq -1350000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 1.8 \cdot 10^{+83}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (- b) (* c z))))
   (if (<= c -1350000000000.0) t_1 (if (<= c 1.8e+83) (* (* i t) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = -b * (c * z);
	double tmp;
	if (c <= -1350000000000.0) {
		tmp = t_1;
	} else if (c <= 1.8e+83) {
		tmp = (i * t) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -b * (c * z)
    if (c <= (-1350000000000.0d0)) then
        tmp = t_1
    else if (c <= 1.8d+83) then
        tmp = (i * t) * b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = -b * (c * z);
	double tmp;
	if (c <= -1350000000000.0) {
		tmp = t_1;
	} else if (c <= 1.8e+83) {
		tmp = (i * t) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = -b * (c * z)
	tmp = 0
	if c <= -1350000000000.0:
		tmp = t_1
	elif c <= 1.8e+83:
		tmp = (i * t) * b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(-b) * Float64(c * z))
	tmp = 0.0
	if (c <= -1350000000000.0)
		tmp = t_1;
	elseif (c <= 1.8e+83)
		tmp = Float64(Float64(i * t) * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = -b * (c * z);
	tmp = 0.0;
	if (c <= -1350000000000.0)
		tmp = t_1;
	elseif (c <= 1.8e+83)
		tmp = (i * t) * b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[((-b) * N[(c * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1350000000000.0], t$95$1, If[LessEqual[c, 1.8e+83], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.35e12 or 1.7999999999999999e83 < c

   1. Initial program 65.6%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

      7. lower-*.f64 46.5

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

   4. Applied rewrites 46.5%

      \[\leadsto \color{blue}{\left(y \cdot x - c \cdot b\right) \cdot z} \]

   5. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\left(b \cdot \left(c \cdot z\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot b\right) \cdot \left(c \cdot \color{blue}{z}\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z\right) \]

      3. lift-neg.f64 N/A

         \[\leadsto \left(-b\right) \cdot \left(c \cdot z\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \left(-b\right) \cdot \left(c \cdot \color{blue}{z}\right) \]

      5. lift-*.f64 36.3

         \[\leadsto \left(-b\right) \cdot \left(c \cdot z\right) \]

   7. Applied rewrites 36.3%

      \[\leadsto \left(-b\right) \cdot \color{blue}{\left(c \cdot z\right)} \]

   if -1.35e12 < c < 1.7999999999999999e83

   1. Initial program 79.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 80.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)} + a \cdot \left(c \cdot j\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1} \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{t \cdot x}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, \color{blue}{t} \cdot x, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, a \cdot \left(c \cdot j\right) + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(c \cdot j\right) \cdot a + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(c \cdot j, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-1 \cdot b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

   6. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right)} \]

   7. Taylor expanded in i around inf

      \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      3. lift-*.f64 25.0

         \[\leadsto \left(i \cdot t\right) \cdot b \]

   9. Applied rewrites 25.0%

      \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 29.3% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{-108}:\\ \;\;\;\;i \cdot \left(b \cdot t\right)\\ \mathbf{elif}\;t \leq 8 \cdot 10^{-129}:\\ \;\;\;\;\left(j \cdot c\right) \cdot a\\ \mathbf{elif}\;t \leq 1.66 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;\left(i \cdot t\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t -1.05e-108)
   (* i (* b t))
   (if (<= t 8e-129)
     (* (* j c) a)
     (if (<= t 1.66e-38) (* y (* x z)) (* (* i t) b)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= -1.05e-108) {
		tmp = i * (b * t);
	} else if (t <= 8e-129) {
		tmp = (j * c) * a;
	} else if (t <= 1.66e-38) {
		tmp = y * (x * z);
	} else {
		tmp = (i * t) * b;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (t <= (-1.05d-108)) then
        tmp = i * (b * t)
    else if (t <= 8d-129) then
        tmp = (j * c) * a
    else if (t <= 1.66d-38) then
        tmp = y * (x * z)
    else
        tmp = (i * t) * b
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= -1.05e-108) {
		tmp = i * (b * t);
	} else if (t <= 8e-129) {
		tmp = (j * c) * a;
	} else if (t <= 1.66e-38) {
		tmp = y * (x * z);
	} else {
		tmp = (i * t) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if t <= -1.05e-108:
		tmp = i * (b * t)
	elif t <= 8e-129:
		tmp = (j * c) * a
	elif t <= 1.66e-38:
		tmp = y * (x * z)
	else:
		tmp = (i * t) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= -1.05e-108)
		tmp = Float64(i * Float64(b * t));
	elseif (t <= 8e-129)
		tmp = Float64(Float64(j * c) * a);
	elseif (t <= 1.66e-38)
		tmp = Float64(y * Float64(x * z));
	else
		tmp = Float64(Float64(i * t) * b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (t <= -1.05e-108)
		tmp = i * (b * t);
	elseif (t <= 8e-129)
		tmp = (j * c) * a;
	elseif (t <= 1.66e-38)
		tmp = y * (x * z);
	else
		tmp = (i * t) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, -1.05e-108], N[(i * N[(b * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 8e-129], N[(N[(j * c), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t, 1.66e-38], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision], N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -1.05e-108

   1. Initial program 70.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 71.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)} + a \cdot \left(c \cdot j\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1} \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{t \cdot x}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, \color{blue}{t} \cdot x, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, a \cdot \left(c \cdot j\right) + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(c \cdot j\right) \cdot a + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(c \cdot j, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-1 \cdot b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

   6. Applied rewrites 63.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right)} \]

   7. Taylor expanded in i around inf

      \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      3. lift-*.f64 28.7

         \[\leadsto \left(i \cdot t\right) \cdot b \]

   9. Applied rewrites 28.7%

      \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(i \cdot t\right) \cdot b \]

       2. lift-*.f64 N/A

          \[\leadsto \left(i \cdot t\right) \cdot b \]

       3. associate-*l* N/A

          \[\leadsto i \cdot \left(t \cdot \color{blue}{b}\right) \]

       4. *-commutative N/A

          \[\leadsto i \cdot \left(b \cdot t\right) \]

       5. lower-*.f64 N/A

          \[\leadsto i \cdot \left(b \cdot \color{blue}{t}\right) \]

       6. lower-*.f64 28.2

          \[\leadsto i \cdot \left(b \cdot t\right) \]

   11. Applied rewrites 28.2%

       \[\leadsto i \cdot \left(b \cdot \color{blue}{t}\right) \]

   if -1.05e-108 < t < 7.9999999999999994e-129

   1. Initial program 79.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{a \cdot \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right) \cdot \color{blue}{a} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(t \cdot x\right) + c \cdot j\right) \cdot \color{blue}{a} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot t\right) \cdot x + c \cdot j\right) \cdot a \]

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(t\right)\right) \cdot x + c \cdot j\right) \cdot a \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(t\right), x, c \cdot j\right) \cdot a \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-t, x, c \cdot j\right) \cdot a \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-t, x, j \cdot c\right) \cdot a \]

      8. lower-*.f64 30.9

         \[\leadsto \mathsf{fma}\left(-t, x, j \cdot c\right) \cdot a \]

   4. Applied rewrites 30.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-t, x, j \cdot c\right) \cdot a} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(c \cdot j\right) \cdot a \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(j \cdot c\right) \cdot a \]

      2. lift-*.f64 25.9

         \[\leadsto \left(j \cdot c\right) \cdot a \]

   7. Applied rewrites 25.9%

      \[\leadsto \left(j \cdot c\right) \cdot a \]

   if 7.9999999999999994e-129 < t < 1.66000000000000006e-38

   1. Initial program 81.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

      7. lower-*.f64 43.7

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

   4. Applied rewrites 43.7%

      \[\leadsto \color{blue}{\left(y \cdot x - c \cdot b\right) \cdot z} \]

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot z\right) \cdot x \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot z\right) \cdot x \]

      3. lower-*.f64 24.6

         \[\leadsto \left(y \cdot z\right) \cdot x \]

   7. Applied rewrites 24.6%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(y \cdot z\right) \cdot x \]

      2. lift-*.f64 N/A

         \[\leadsto \left(y \cdot z\right) \cdot x \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \left(z \cdot \color{blue}{x}\right) \]

      4. *-commutative N/A

         \[\leadsto y \cdot \left(x \cdot z\right) \]

      5. lower-*.f64 N/A

         \[\leadsto y \cdot \left(x \cdot \color{blue}{z}\right) \]

      6. lower-*.f64 25.6

         \[\leadsto y \cdot \left(x \cdot z\right) \]

   9. Applied rewrites 25.6%

      \[\leadsto y \cdot \left(x \cdot \color{blue}{z}\right) \]

   if 1.66000000000000006e-38 < t

   1. Initial program 68.8%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 70.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)} + a \cdot \left(c \cdot j\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1} \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{t \cdot x}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, \color{blue}{t} \cdot x, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, a \cdot \left(c \cdot j\right) + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(c \cdot j\right) \cdot a + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(c \cdot j, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-1 \cdot b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

   6. Applied rewrites 65.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right)} \]

   7. Taylor expanded in i around inf

      \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      3. lift-*.f64 31.7

         \[\leadsto \left(i \cdot t\right) \cdot b \]

   9. Applied rewrites 31.7%

      \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 17: 29.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+91}:\\ \;\;\;\;\left(y \cdot z\right) \cdot x\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -5.5e+91)
   (* (* y z) x)
   (if (<= z 1.15e+20) (* (* i b) t) (* y (* x z)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -5.5e+91) {
		tmp = (y * z) * x;
	} else if (z <= 1.15e+20) {
		tmp = (i * b) * t;
	} else {
		tmp = y * (x * z);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if (z <= (-5.5d+91)) then
        tmp = (y * z) * x
    else if (z <= 1.15d+20) then
        tmp = (i * b) * t
    else
        tmp = y * (x * z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (z <= -5.5e+91) {
		tmp = (y * z) * x;
	} else if (z <= 1.15e+20) {
		tmp = (i * b) * t;
	} else {
		tmp = y * (x * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -5.5e+91:
		tmp = (y * z) * x
	elif z <= 1.15e+20:
		tmp = (i * b) * t
	else:
		tmp = y * (x * z)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -5.5e+91)
		tmp = Float64(Float64(y * z) * x);
	elseif (z <= 1.15e+20)
		tmp = Float64(Float64(i * b) * t);
	else
		tmp = Float64(y * Float64(x * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -5.5e+91)
		tmp = (y * z) * x;
	elseif (z <= 1.15e+20)
		tmp = (i * b) * t;
	else
		tmp = y * (x * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -5.5e+91], N[(N[(y * z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.15e+20], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4999999999999998e91

   1. Initial program 62.2%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

      7. lower-*.f64 66.9

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

   4. Applied rewrites 66.9%

      \[\leadsto \color{blue}{\left(y \cdot x - c \cdot b\right) \cdot z} \]

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot z\right) \cdot x \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot z\right) \cdot x \]

      3. lower-*.f64 38.9

         \[\leadsto \left(y \cdot z\right) \cdot x \]

   7. Applied rewrites 38.9%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]

   if -5.4999999999999998e91 < z < 1.15e20

   1. Initial program 79.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)} + a \cdot \left(c \cdot j\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1} \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{t \cdot x}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, \color{blue}{t} \cdot x, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, a \cdot \left(c \cdot j\right) + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(c \cdot j\right) \cdot a + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(c \cdot j, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-1 \cdot b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

   6. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right)} \]

   7. Taylor expanded in i around inf

      \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      3. lift-*.f64 25.5

         \[\leadsto \left(i \cdot t\right) \cdot b \]

   9. Applied rewrites 25.5%

      \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(i \cdot t\right) \cdot b \]

       2. lift-*.f64 N/A

          \[\leadsto \left(i \cdot t\right) \cdot b \]

       3. *-commutative N/A

          \[\leadsto b \cdot \left(i \cdot \color{blue}{t}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(b \cdot i\right) \cdot t \]

       5. lower-*.f64 N/A

          \[\leadsto \left(b \cdot i\right) \cdot t \]

       6. *-commutative N/A

          \[\leadsto \left(i \cdot b\right) \cdot t \]

       7. lower-*.f64 24.7

          \[\leadsto \left(i \cdot b\right) \cdot t \]

   11. Applied rewrites 24.7%

       \[\leadsto \left(i \cdot b\right) \cdot t \]

   if 1.15e20 < z

   1. Initial program 67.3%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

      7. lower-*.f64 60.1

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

   4. Applied rewrites 60.1%

      \[\leadsto \color{blue}{\left(y \cdot x - c \cdot b\right) \cdot z} \]

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot z\right) \cdot x \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot z\right) \cdot x \]

      3. lower-*.f64 35.3

         \[\leadsto \left(y \cdot z\right) \cdot x \]

   7. Applied rewrites 35.3%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(y \cdot z\right) \cdot x \]

      2. lift-*.f64 N/A

         \[\leadsto \left(y \cdot z\right) \cdot x \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \left(z \cdot \color{blue}{x}\right) \]

      4. *-commutative N/A

         \[\leadsto y \cdot \left(x \cdot z\right) \]

      5. lower-*.f64 N/A

         \[\leadsto y \cdot \left(x \cdot \color{blue}{z}\right) \]

      6. lower-*.f64 34.6

         \[\leadsto y \cdot \left(x \cdot z\right) \]

   9. Applied rewrites 34.6%

      \[\leadsto y \cdot \left(x \cdot \color{blue}{z}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 18: 28.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(x \cdot z\right)\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+20}:\\ \;\;\;\;\left(i \cdot b\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* y (* x z))))
   (if (<= z -5.5e+91) t_1 (if (<= z 1.15e+20) (* (* i b) t) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = y * (x * z);
	double tmp;
	if (z <= -5.5e+91) {
		tmp = t_1;
	} else if (z <= 1.15e+20) {
		tmp = (i * b) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (x * z)
    if (z <= (-5.5d+91)) then
        tmp = t_1
    else if (z <= 1.15d+20) then
        tmp = (i * b) * t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = y * (x * z);
	double tmp;
	if (z <= -5.5e+91) {
		tmp = t_1;
	} else if (z <= 1.15e+20) {
		tmp = (i * b) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = y * (x * z)
	tmp = 0
	if z <= -5.5e+91:
		tmp = t_1
	elif z <= 1.15e+20:
		tmp = (i * b) * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(y * Float64(x * z))
	tmp = 0.0
	if (z <= -5.5e+91)
		tmp = t_1;
	elseif (z <= 1.15e+20)
		tmp = Float64(Float64(i * b) * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = y * (x * z);
	tmp = 0.0;
	if (z <= -5.5e+91)
		tmp = t_1;
	elseif (z <= 1.15e+20)
		tmp = (i * b) * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(y * N[(x * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5e+91], t$95$1, If[LessEqual[z, 1.15e+20], N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4999999999999998e91 or 1.15e20 < z

   1. Initial program 65.1%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{z \cdot \left(x \cdot y - b \cdot c\right)} \]
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot \color{blue}{z} \]

      3. lower--.f64 N/A

         \[\leadsto \left(x \cdot y - b \cdot c\right) \cdot z \]

      4. *-commutative N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      5. lower-*.f64 N/A

         \[\leadsto \left(y \cdot x - b \cdot c\right) \cdot z \]

      6. *-commutative N/A

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

      7. lower-*.f64 63.1

         \[\leadsto \left(y \cdot x - c \cdot b\right) \cdot z \]

   4. Applied rewrites 63.1%

      \[\leadsto \color{blue}{\left(y \cdot x - c \cdot b\right) \cdot z} \]

   5. Taylor expanded in x around inf

      \[\leadsto x \cdot \color{blue}{\left(y \cdot z\right)} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(y \cdot z\right) \cdot x \]

      2. lower-*.f64 N/A

         \[\leadsto \left(y \cdot z\right) \cdot x \]

      3. lower-*.f64 36.9

         \[\leadsto \left(y \cdot z\right) \cdot x \]

   7. Applied rewrites 36.9%

      \[\leadsto \left(y \cdot z\right) \cdot \color{blue}{x} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left(y \cdot z\right) \cdot x \]

      2. lift-*.f64 N/A

         \[\leadsto \left(y \cdot z\right) \cdot x \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \left(z \cdot \color{blue}{x}\right) \]

      4. *-commutative N/A

         \[\leadsto y \cdot \left(x \cdot z\right) \]

      5. lower-*.f64 N/A

         \[\leadsto y \cdot \left(x \cdot \color{blue}{z}\right) \]

      6. lower-*.f64 36.5

         \[\leadsto y \cdot \left(x \cdot z\right) \]

   9. Applied rewrites 36.5%

      \[\leadsto y \cdot \left(x \cdot \color{blue}{z}\right) \]

   if -5.4999999999999998e91 < z < 1.15e20

   1. Initial program 79.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      2. lift-*.f64 N/A

         \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      3. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      4. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      5. lift--.f64 N/A

         \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      7. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      9. lift--.f64 N/A

         \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      10. fp-cancel-sub-sign-inv N/A

          \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      11. *-commutative N/A

          \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      12. *-commutative N/A

          \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      13. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

      14. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      17. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

      18. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. Applied rewrites 80.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot a\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)} + a \cdot \left(c \cdot j\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1} \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{t \cdot x}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      4. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, \color{blue}{t} \cdot x, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, a \cdot \left(c \cdot j\right) + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(c \cdot j\right) \cdot a + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      9. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(c \cdot j, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      11. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

      12. associate-*r* N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-1 \cdot b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      13. mul-1-neg N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      14. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

      15. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

   6. Applied rewrites 64.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right)} \]

   7. Taylor expanded in i around inf

      \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
   8. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      2. lower-*.f64 N/A

         \[\leadsto \left(i \cdot t\right) \cdot b \]

      3. lift-*.f64 25.5

         \[\leadsto \left(i \cdot t\right) \cdot b \]

   9. Applied rewrites 25.5%

      \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]
   10. Step-by-step derivation

       1. lift-*.f64 N/A

          \[\leadsto \left(i \cdot t\right) \cdot b \]

       2. lift-*.f64 N/A

          \[\leadsto \left(i \cdot t\right) \cdot b \]

       3. *-commutative N/A

          \[\leadsto b \cdot \left(i \cdot \color{blue}{t}\right) \]

       4. associate-*r* N/A

          \[\leadsto \left(b \cdot i\right) \cdot t \]

       5. lower-*.f64 N/A

          \[\leadsto \left(b \cdot i\right) \cdot t \]

       6. *-commutative N/A

          \[\leadsto \left(i \cdot b\right) \cdot t \]

       7. lower-*.f64 24.7

          \[\leadsto \left(i \cdot b\right) \cdot t \]

   11. Applied rewrites 24.7%

       \[\leadsto \left(i \cdot b\right) \cdot t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 19: 21.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \left(i \cdot t\right) \cdot b \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* i t) b))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (i * t) * b;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = (i * t) * b
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (i * t) * b;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (i * t) * b

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(i * t) * b)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (i * t) * b;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(i * t), $MachinePrecision] * b), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   5. lift--.f64 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   9. lift--.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   10. fp-cancel-sub-sign-inv N/A

       \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   11. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   12. *-commutative N/A

       \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   14. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

3. Applied rewrites 74.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

4. Taylor expanded in y around 0

   \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right)} \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot a\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)} + a \cdot \left(c \cdot j\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1} \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{t \cdot x}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

   4. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(-a, \color{blue}{t} \cdot x, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(-a, x \cdot t, a \cdot \left(c \cdot j\right) + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(c \cdot j\right) \cdot a + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(c \cdot j, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-1 \cdot b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

   13. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

   14. lift-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

6. Applied rewrites 60.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right)} \]

7. Taylor expanded in i around inf

   \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(i \cdot t\right) \cdot b \]

   2. lower-*.f64 N/A

      \[\leadsto \left(i \cdot t\right) \cdot b \]

   3. lift-*.f64 21.7

      \[\leadsto \left(i \cdot t\right) \cdot b \]

9. Applied rewrites 21.7%

   \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]
10. Add Preprocessing

Alternative 20: 21.7% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \left(i \cdot b\right) \cdot t \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* i b) t))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (i * b) * t;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = (i * b) * t
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (i * b) * t;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (i * b) * t

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(i * b) * t)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (i * b) * t;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(i * b), $MachinePrecision] * t), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   5. lift--.f64 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   9. lift--.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   10. fp-cancel-sub-sign-inv N/A

       \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   11. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   12. *-commutative N/A

       \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   14. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

3. Applied rewrites 74.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

4. Taylor expanded in y around 0

   \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right)} \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot a\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)} + a \cdot \left(c \cdot j\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1} \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{t \cdot x}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

   4. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(-a, \color{blue}{t} \cdot x, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(-a, x \cdot t, a \cdot \left(c \cdot j\right) + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(c \cdot j\right) \cdot a + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(c \cdot j, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-1 \cdot b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

   13. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

   14. lift-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

6. Applied rewrites 60.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right)} \]

7. Taylor expanded in i around inf

   \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(i \cdot t\right) \cdot b \]

   2. lower-*.f64 N/A

      \[\leadsto \left(i \cdot t\right) \cdot b \]

   3. lift-*.f64 21.7

      \[\leadsto \left(i \cdot t\right) \cdot b \]

9. Applied rewrites 21.7%

   \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(i \cdot t\right) \cdot b \]

    2. lift-*.f64 N/A

       \[\leadsto \left(i \cdot t\right) \cdot b \]

    3. *-commutative N/A

       \[\leadsto b \cdot \left(i \cdot \color{blue}{t}\right) \]

    4. associate-*r* N/A

       \[\leadsto \left(b \cdot i\right) \cdot t \]

    5. lower-*.f64 N/A

       \[\leadsto \left(b \cdot i\right) \cdot t \]

    6. *-commutative N/A

       \[\leadsto \left(i \cdot b\right) \cdot t \]

    7. lower-*.f64 21.5

       \[\leadsto \left(i \cdot b\right) \cdot t \]

11. Applied rewrites 21.5%

    \[\leadsto \left(i \cdot b\right) \cdot t \]
12. Add Preprocessing

Alternative 21: 21.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ i \cdot \left(b \cdot t\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* i (* b t)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return i * (b * t);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    code = i * (b * t)
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return i * (b * t);
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return i * (b * t)

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(i * Float64(b * t))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = i * (b * t);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(i * N[(b * t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.5%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{x \cdot \left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \left(\color{blue}{y \cdot z} - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   4. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot z - \color{blue}{t \cdot a}\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   5. lift--.f64 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(y \cdot z - t \cdot a\right)} - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   6. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - \color{blue}{b \cdot \left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   7. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(\color{blue}{c \cdot z} - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - \color{blue}{t \cdot i}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   9. lift--.f64 N/A

      \[\leadsto \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \color{blue}{\left(c \cdot z - t \cdot i\right)}\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   10. fp-cancel-sub-sign-inv N/A

       \[\leadsto \color{blue}{\left(x \cdot \left(y \cdot z - t \cdot a\right) + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   11. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(y \cdot z - t \cdot a\right) \cdot x} + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   12. *-commutative N/A

       \[\leadsto \left(\left(y \cdot z - t \cdot a\right) \cdot x + \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - \color{blue}{i \cdot t}\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

   14. lift--.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z - t \cdot a}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y} - t \cdot a, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   17. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(z \cdot y - \color{blue}{a \cdot t}, x, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]

3. Applied rewrites 74.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y - a \cdot t, x, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)} + j \cdot \left(c \cdot a - y \cdot i\right) \]

4. Taylor expanded in y around 0

   \[\leadsto \color{blue}{-1 \cdot \left(a \cdot \left(t \cdot x\right)\right) + \left(-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right)} \]
5. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot a\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)} + a \cdot \left(c \cdot j\right)\right) \]

   2. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(a\right)\right) \cdot \left(t \cdot x\right) + \left(\color{blue}{-1} \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

   3. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), \color{blue}{t \cdot x}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

   4. lower-neg.f64 N/A

      \[\leadsto \mathsf{fma}\left(-a, \color{blue}{t} \cdot x, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-a, x \cdot \color{blue}{t}, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right) + a \cdot \left(c \cdot j\right)\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(-a, x \cdot t, a \cdot \left(c \cdot j\right) + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \left(c \cdot j\right) \cdot a + -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

   9. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(c \cdot j, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, -1 \cdot \left(b \cdot \left(c \cdot z - i \cdot t\right)\right)\right)\right) \]

   12. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-1 \cdot b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

   13. mul-1-neg N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(\mathsf{neg}\left(b\right)\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

   14. lift-neg.f64 N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(-b\right) \cdot \left(c \cdot z - i \cdot t\right)\right)\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right) \]

6. Applied rewrites 60.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x \cdot t, \mathsf{fma}\left(j \cdot c, a, \left(c \cdot z - i \cdot t\right) \cdot \left(-b\right)\right)\right)} \]

7. Taylor expanded in i around inf

   \[\leadsto b \cdot \color{blue}{\left(i \cdot t\right)} \]
8. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(i \cdot t\right) \cdot b \]

   2. lower-*.f64 N/A

      \[\leadsto \left(i \cdot t\right) \cdot b \]

   3. lift-*.f64 21.7

      \[\leadsto \left(i \cdot t\right) \cdot b \]

9. Applied rewrites 21.7%

   \[\leadsto \left(i \cdot t\right) \cdot \color{blue}{b} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \left(i \cdot t\right) \cdot b \]

    2. lift-*.f64 N/A

       \[\leadsto \left(i \cdot t\right) \cdot b \]

    3. associate-*l* N/A

       \[\leadsto i \cdot \left(t \cdot \color{blue}{b}\right) \]

    4. *-commutative N/A

       \[\leadsto i \cdot \left(b \cdot t\right) \]

    5. lower-*.f64 N/A

       \[\leadsto i \cdot \left(b \cdot \color{blue}{t}\right) \]

    6. lower-*.f64 21.7

       \[\leadsto i \cdot \left(b \cdot t\right) \]

11. Applied rewrites 21.7%

    \[\leadsto i \cdot \left(b \cdot \color{blue}{t}\right) \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025119 
(FPCore (x y z t a b c i j)
  :name "Data.Colour.Matrix:determinant from colour-2.3.3, A"
  :precision binary64
  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i)))) (* j (- (* c a) (* y i)))))