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

Percentage Accurate: 72.9% → 81.1%

Time: 8.3s

Alternatives: 19

Speedup: 0.5×

• 58.2% 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: 72.9% 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: 81.1% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<=
      (+
       (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* t i))))
       (* j (- (* c a) (* y i))))
      INFINITY)
   (fma
    (- (* c a) (* i y))
    j
    (fma (- (* i t) (* c z)) b (* (- (* z y) (* a t)) x)))
   (* c (- (* a j) (* b 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 ((((x * ((y * z) - (t * a))) - (b * ((c * z) - (t * i)))) + (j * ((c * a) - (y * i)))) <= ((double) INFINITY)) {
		tmp = fma(((c * a) - (i * y)), j, fma(((i * t) - (c * z)), b, (((z * y) - (a * t)) * x)));
	} else {
		tmp = c * ((a * j) - (b * z));
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := 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] + N[(j * N[(N[(c * a), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j + N[(N[(N[(i * t), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision] * b + N[(N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(a * j), $MachinePrecision] - N[(b * z), $MachinePrecision]), $MachinePrecision]), $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 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

   3. Applied rewrites 76.0%

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

   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 72.9%

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

   2. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 38.7

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

   4. Applied rewrites 38.7%

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

Alternative 2: 70.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot a - i \cdot y\\ \mathbf{if}\;j \leq -1.6 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, j, x \cdot \left(y \cdot z\right)\right)\\ \mathbf{elif}\;j \leq 2.2 \cdot 10^{-80}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y - t \cdot a, x, \mathsf{fma}\left(b, t \cdot i - z \cdot c, j \cdot \left(a \cdot c\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, j, b \cdot \left(i \cdot t - c \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (- (* c a) (* i y))))
   (if (<= j -1.6e+88)
     (fma t_1 j (* x (* y z)))
     (if (<= j 2.2e-80)
       (fma (- (* z y) (* t a)) x (fma b (- (* t i) (* z c)) (* j (* a c))))
       (fma t_1 j (* b (- (* i t) (* c z))))))))

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);
	double tmp;
	if (j <= -1.6e+88) {
		tmp = fma(t_1, j, (x * (y * z)));
	} else if (j <= 2.2e-80) {
		tmp = fma(((z * y) - (t * a)), x, fma(b, ((t * i) - (z * c)), (j * (a * c))));
	} else {
		tmp = fma(t_1, j, (b * ((i * t) - (c * z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(c * a) - Float64(i * y))
	tmp = 0.0
	if (j <= -1.6e+88)
		tmp = fma(t_1, j, Float64(x * Float64(y * z)));
	elseif (j <= 2.2e-80)
		tmp = fma(Float64(Float64(z * y) - Float64(t * a)), x, fma(b, Float64(Float64(t * i) - Float64(z * c)), Float64(j * Float64(a * c))));
	else
		tmp = fma(t_1, j, Float64(b * Float64(Float64(i * t) - Float64(c * z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -1.6e+88], N[(t$95$1 * j + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[j, 2.2e-80], N[(N[(N[(z * y), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision] * x + N[(b * N[(N[(t * i), $MachinePrecision] - N[(z * c), $MachinePrecision]), $MachinePrecision] + N[(j * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * j + N[(b * N[(N[(i * t), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if j < -1.5999999999999999e88

   1. Initial program 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

   3. Applied rewrites 76.0%

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

   4. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(c \cdot a - i \cdot y, j, \color{blue}{x \cdot \left(y \cdot z\right)}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 49.7

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

   6. Applied rewrites 49.7%

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

   if -1.5999999999999999e88 < j < 2.2000000000000001e-80

   1. Initial program 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

   3. Applied rewrites 76.0%

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 67.7

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

   8. Applied rewrites 67.7%

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

   if 2.2000000000000001e-80 < j

   1. Initial program 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

   3. Applied rewrites 76.0%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 60.7

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

   6. Applied rewrites 60.7%

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

Alternative 3: 69.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(i \cdot t - c \cdot z\right)\\ \mathbf{if}\;b \leq -3.2 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot a - i \cdot y, j, t\_1\right)\\ \mathbf{elif}\;b \leq -3.7 \cdot 10^{-54}:\\ \;\;\;\;\mathsf{fma}\left(a, c \cdot j, x \cdot \left(y \cdot z - a \cdot t\right)\right)\\ \mathbf{elif}\;b \leq 3.15 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot z, y, \mathsf{fma}\left(\left(-x\right) \cdot a, t, j \cdot \left(a \cdot c - y \cdot i\right)\right)\right)\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(j, a \cdot c - i \cdot y, x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (- (* i t) (* c z)))))
   (if (<= b -3.2e+91)
     (fma (- (* c a) (* i y)) j t_1)
     (if (<= b -3.7e-54)
       (fma a (* c j) (* x (- (* y z) (* a t))))
       (if (<= b 3.15e-27)
         (fma (* x z) y (fma (* (- x) a) t (* j (- (* a c) (* y i)))))
         (if (<= b 7.2e+76)
           (- (fma j (- (* a c) (* i y)) (* x (* y z))) (* 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 = b * ((i * t) - (c * z));
	double tmp;
	if (b <= -3.2e+91) {
		tmp = fma(((c * a) - (i * y)), j, t_1);
	} else if (b <= -3.7e-54) {
		tmp = fma(a, (c * j), (x * ((y * z) - (a * t))));
	} else if (b <= 3.15e-27) {
		tmp = fma((x * z), y, fma((-x * a), t, (j * ((a * c) - (y * i)))));
	} else if (b <= 7.2e+76) {
		tmp = fma(j, ((a * c) - (i * y)), (x * (y * z))) - (b * (c * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * Float64(Float64(i * t) - Float64(c * z)))
	tmp = 0.0
	if (b <= -3.2e+91)
		tmp = fma(Float64(Float64(c * a) - Float64(i * y)), j, t_1);
	elseif (b <= -3.7e-54)
		tmp = fma(a, Float64(c * j), Float64(x * Float64(Float64(y * z) - Float64(a * t))));
	elseif (b <= 3.15e-27)
		tmp = fma(Float64(x * z), y, fma(Float64(Float64(-x) * a), t, Float64(j * Float64(Float64(a * c) - Float64(y * i)))));
	elseif (b <= 7.2e+76)
		tmp = Float64(fma(j, Float64(Float64(a * c) - Float64(i * y)), Float64(x * Float64(y * z))) - 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[(b * N[(N[(i * t), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.2e+91], N[(N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j + t$95$1), $MachinePrecision], If[LessEqual[b, -3.7e-54], N[(a * N[(c * j), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.15e-27], N[(N[(x * z), $MachinePrecision] * y + N[(N[((-x) * a), $MachinePrecision] * t + N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e+76], N[(N[(j * N[(N[(a * c), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -3.19999999999999989e91

   1. Initial program 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

   3. Applied rewrites 76.0%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 60.7

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

   6. Applied rewrites 60.7%

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

   if -3.19999999999999989e91 < b < -3.7000000000000003e-54

   1. Initial program 72.9%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 60.3

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in i around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 50.8

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

   7. Applied rewrites 50.8%

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

   if -3.7000000000000003e-54 < b < 3.15000000000000005e-27

   1. Initial program 72.9%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 60.3

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

   4. Applied rewrites 60.3%

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

   6. Applied rewrites 59.4%

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

   if 3.15000000000000005e-27 < b < 7.2000000000000006e76

   1. Initial program 72.9%

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

   2. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 56.4

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

   4. Applied rewrites 56.4%

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

   if 7.2000000000000006e76 < b

   1. Initial program 72.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.3

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

   4. Applied rewrites 39.3%

      \[\leadsto \color{blue}{b \cdot \left(i \cdot t - c \cdot z\right)} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 4: 69.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(i \cdot t - c \cdot z\right)\\ \mathbf{if}\;b \leq -5.6 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot a - i \cdot y, j, t\_1\right)\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y - t \cdot a, x, j \cdot \left(a \cdot c - y \cdot i\right)\right)\\ \mathbf{elif}\;b \leq 7.2 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(j, a \cdot c - i \cdot y, x \cdot \left(y \cdot z\right)\right) - b \cdot \left(c \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (- (* i t) (* c z)))))
   (if (<= b -5.6e+91)
     (fma (- (* c a) (* i y)) j t_1)
     (if (<= b 1.45e-27)
       (fma (- (* z y) (* t a)) x (* j (- (* a c) (* y i))))
       (if (<= b 7.2e+76)
         (- (fma j (- (* a c) (* i y)) (* x (* y z))) (* 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 = b * ((i * t) - (c * z));
	double tmp;
	if (b <= -5.6e+91) {
		tmp = fma(((c * a) - (i * y)), j, t_1);
	} else if (b <= 1.45e-27) {
		tmp = fma(((z * y) - (t * a)), x, (j * ((a * c) - (y * i))));
	} else if (b <= 7.2e+76) {
		tmp = fma(j, ((a * c) - (i * y)), (x * (y * z))) - (b * (c * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * Float64(Float64(i * t) - Float64(c * z)))
	tmp = 0.0
	if (b <= -5.6e+91)
		tmp = fma(Float64(Float64(c * a) - Float64(i * y)), j, t_1);
	elseif (b <= 1.45e-27)
		tmp = fma(Float64(Float64(z * y) - Float64(t * a)), x, Float64(j * Float64(Float64(a * c) - Float64(y * i))));
	elseif (b <= 7.2e+76)
		tmp = Float64(fma(j, Float64(Float64(a * c) - Float64(i * y)), Float64(x * Float64(y * z))) - 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[(b * N[(N[(i * t), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5.6e+91], N[(N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j + t$95$1), $MachinePrecision], If[LessEqual[b, 1.45e-27], N[(N[(N[(z * y), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision] * x + N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.2e+76], N[(N[(j * N[(N[(a * c), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 4 regimes

2. if b < -5.5999999999999997e91

   1. Initial program 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

   3. Applied rewrites 76.0%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 60.7

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

   6. Applied rewrites 60.7%

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

   if -5.5999999999999997e91 < b < 1.45000000000000002e-27

   1. Initial program 72.9%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 60.3

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

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, a \cdot c - i \cdot y, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 60.2

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

   6. Applied rewrites 60.2%

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

   if 1.45000000000000002e-27 < b < 7.2000000000000006e76

   1. Initial program 72.9%

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

   2. Taylor expanded in t around 0

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

      1. lower--.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lower-*.f64 56.4

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

   4. Applied rewrites 56.4%

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

   if 7.2000000000000006e76 < b

   1. Initial program 72.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.3

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

   4. Applied rewrites 39.3%

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

Alternative 5: 67.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(c \cdot a - i \cdot y, j, b \cdot \left(i \cdot t - c \cdot z\right)\right)\\ \mathbf{if}\;b \leq -5.6 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-104}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y - t \cdot a, x, j \cdot \left(a \cdot c - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (fma (- (* c a) (* i y)) j (* b (- (* i t) (* c z))))))
   (if (<= b -5.6e+91)
     t_1
     (if (<= b 8.2e-104)
       (fma (- (* z y) (* t a)) x (* j (- (* a c) (* y i))))
       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 = fma(((c * a) - (i * y)), j, (b * ((i * t) - (c * z))));
	double tmp;
	if (b <= -5.6e+91) {
		tmp = t_1;
	} else if (b <= 8.2e-104) {
		tmp = fma(((z * y) - (t * a)), x, (j * ((a * c) - (y * i))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = fma(Float64(Float64(c * a) - Float64(i * y)), j, Float64(b * Float64(Float64(i * t) - Float64(c * z))))
	tmp = 0.0
	if (b <= -5.6e+91)
		tmp = t_1;
	elseif (b <= 8.2e-104)
		tmp = fma(Float64(Float64(z * y) - Float64(t * a)), x, Float64(j * Float64(Float64(a * c) - Float64(y * i))));
	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 + N[(b * N[(N[(i * t), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -5.6e+91], t$95$1, If[LessEqual[b, 8.2e-104], N[(N[(N[(z * y), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision] * x + N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -5.5999999999999997e91 or 8.19999999999999968e-104 < b

   1. Initial program 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

   3. Applied rewrites 76.0%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 60.7

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

   6. Applied rewrites 60.7%

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

   if -5.5999999999999997e91 < b < 8.19999999999999968e-104

   1. Initial program 72.9%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 60.3

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

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, a \cdot c - i \cdot y, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 60.2

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

   6. Applied rewrites 60.2%

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

Alternative 6: 67.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(i \cdot t - c \cdot z\right)\\ \mathbf{if}\;b \leq -8.5 \cdot 10^{+130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2.5 \cdot 10^{+77}:\\ \;\;\;\;\mathsf{fma}\left(z \cdot y - t \cdot a, x, j \cdot \left(a \cdot c - y \cdot i\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (- (* i t) (* c z)))))
   (if (<= b -8.5e+130)
     t_1
     (if (<= b 2.5e+77)
       (fma (- (* z y) (* t a)) x (* j (- (* a c) (* y i))))
       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 * ((i * t) - (c * z));
	double tmp;
	if (b <= -8.5e+130) {
		tmp = t_1;
	} else if (b <= 2.5e+77) {
		tmp = fma(((z * y) - (t * a)), x, (j * ((a * c) - (y * i))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * Float64(Float64(i * t) - Float64(c * z)))
	tmp = 0.0
	if (b <= -8.5e+130)
		tmp = t_1;
	elseif (b <= 2.5e+77)
		tmp = fma(Float64(Float64(z * y) - Float64(t * a)), x, Float64(j * Float64(Float64(a * c) - Float64(y * i))));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(N[(i * t), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -8.5e+130], t$95$1, If[LessEqual[b, 2.5e+77], N[(N[(N[(z * y), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision] * x + N[(j * N[(N[(a * c), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -8.49999999999999965e130 or 2.50000000000000002e77 < b

   1. Initial program 72.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.3

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

   4. Applied rewrites 39.3%

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

   if -8.49999999999999965e130 < b < 2.50000000000000002e77

   1. Initial program 72.9%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 60.3

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

   4. Applied rewrites 60.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(j, a \cdot c - i \cdot y, x \cdot \left(y \cdot z - a \cdot t\right)\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. lift-*.f64 N/A

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

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

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. lift-*.f64 N/A

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

      13. lift--.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-fma.f64 N/A

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

      16. lift-*.f64 N/A

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

      17. *-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lift-*.f64 N/A

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

      20. *-commutative N/A

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

      21. lower-*.f64 N/A

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

      22. *-commutative N/A

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

      23. lower-*.f64 60.2

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

   6. Applied rewrites 60.2%

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

Alternative 7: 67.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(i \cdot t - c \cdot z\right)\\ \mathbf{if}\;b \leq -3.4 \cdot 10^{+128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(j, a \cdot c - i \cdot y, x \cdot \left(y \cdot z - a \cdot t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (- (* i t) (* c z)))))
   (if (<= b -3.4e+128)
     t_1
     (if (<= b 2e+62)
       (fma j (- (* a c) (* i y)) (* x (- (* y z) (* a 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 = b * ((i * t) - (c * z));
	double tmp;
	if (b <= -3.4e+128) {
		tmp = t_1;
	} else if (b <= 2e+62) {
		tmp = fma(j, ((a * c) - (i * y)), (x * ((y * z) - (a * t))));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * Float64(Float64(i * t) - Float64(c * z)))
	tmp = 0.0
	if (b <= -3.4e+128)
		tmp = t_1;
	elseif (b <= 2e+62)
		tmp = fma(j, Float64(Float64(a * c) - Float64(i * y)), Float64(x * Float64(Float64(y * z) - Float64(a * 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[(b * N[(N[(i * t), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.4e+128], t$95$1, If[LessEqual[b, 2e+62], N[(j * N[(N[(a * c), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if b < -3.3999999999999999e128 or 2.00000000000000007e62 < b

   1. Initial program 72.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.3

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

   4. Applied rewrites 39.3%

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

   if -3.3999999999999999e128 < b < 2.00000000000000007e62

   1. Initial program 72.9%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 60.3

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

   4. Applied rewrites 60.3%

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

Alternative 8: 60.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(i \cdot t - c \cdot z\right)\\ \mathbf{if}\;b \leq -3.3 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -8.5 \cdot 10^{-223}:\\ \;\;\;\;\mathsf{fma}\left(a, c \cdot j, x \cdot \left(y \cdot z - a \cdot t\right)\right)\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(c \cdot a - i \cdot y, j, x \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (- (* i t) (* c z)))))
   (if (<= b -3.3e+91)
     t_1
     (if (<= b -8.5e-223)
       (fma a (* c j) (* x (- (* y z) (* a t))))
       (if (<= b 2e+62) (fma (- (* c a) (* i y)) j (* x (* y 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 = b * ((i * t) - (c * z));
	double tmp;
	if (b <= -3.3e+91) {
		tmp = t_1;
	} else if (b <= -8.5e-223) {
		tmp = fma(a, (c * j), (x * ((y * z) - (a * t))));
	} else if (b <= 2e+62) {
		tmp = fma(((c * a) - (i * y)), j, (x * (y * z)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * Float64(Float64(i * t) - Float64(c * z)))
	tmp = 0.0
	if (b <= -3.3e+91)
		tmp = t_1;
	elseif (b <= -8.5e-223)
		tmp = fma(a, Float64(c * j), Float64(x * Float64(Float64(y * z) - Float64(a * t))));
	elseif (b <= 2e+62)
		tmp = fma(Float64(Float64(c * a) - Float64(i * y)), j, Float64(x * Float64(y * 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[(b * N[(N[(i * t), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.3e+91], t$95$1, If[LessEqual[b, -8.5e-223], N[(a * N[(c * j), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+62], N[(N[(N[(c * a), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision] * j + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.30000000000000017e91 or 2.00000000000000007e62 < b

   1. Initial program 72.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.3

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

   4. Applied rewrites 39.3%

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

   if -3.30000000000000017e91 < b < -8.5000000000000003e-223

   1. Initial program 72.9%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 60.3

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in i around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 50.8

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

   7. Applied rewrites 50.8%

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

   if -8.5000000000000003e-223 < b < 2.00000000000000007e62

   1. Initial program 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

   3. Applied rewrites 76.0%

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

   4. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(c \cdot a - i \cdot y, j, \color{blue}{x \cdot \left(y \cdot z\right)}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-*.f64 49.7

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

   6. Applied rewrites 49.7%

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

Alternative 9: 56.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(i \cdot t - c \cdot z\right)\\ \mathbf{if}\;b \leq -3.3 \cdot 10^{+91}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-266}:\\ \;\;\;\;\mathsf{fma}\left(a, c \cdot j, x \cdot \left(y \cdot z - a \cdot t\right)\right)\\ \mathbf{elif}\;b \leq 10^{-30}:\\ \;\;\;\;j \cdot \left(a \cdot c - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (- (* i t) (* c z)))))
   (if (<= b -3.3e+91)
     t_1
     (if (<= b 3.2e-266)
       (fma a (* c j) (* x (- (* y z) (* a t))))
       (if (<= b 1e-30) (* j (- (* a c) (* i y))) 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 * ((i * t) - (c * z));
	double tmp;
	if (b <= -3.3e+91) {
		tmp = t_1;
	} else if (b <= 3.2e-266) {
		tmp = fma(a, (c * j), (x * ((y * z) - (a * t))));
	} else if (b <= 1e-30) {
		tmp = j * ((a * c) - (i * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * Float64(Float64(i * t) - Float64(c * z)))
	tmp = 0.0
	if (b <= -3.3e+91)
		tmp = t_1;
	elseif (b <= 3.2e-266)
		tmp = fma(a, Float64(c * j), Float64(x * Float64(Float64(y * z) - Float64(a * t))));
	elseif (b <= 1e-30)
		tmp = Float64(j * Float64(Float64(a * c) - Float64(i * y)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(b * N[(N[(i * t), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -3.3e+91], t$95$1, If[LessEqual[b, 3.2e-266], N[(a * N[(c * j), $MachinePrecision] + N[(x * N[(N[(y * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-30], N[(j * N[(N[(a * c), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.30000000000000017e91 or 1e-30 < b

   1. Initial program 72.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.3

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

   4. Applied rewrites 39.3%

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

   if -3.30000000000000017e91 < b < 3.2e-266

   1. Initial program 72.9%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 60.3

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in i around 0

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

      1. lower-fma.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 50.8

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

   7. Applied rewrites 50.8%

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

   if 3.2e-266 < b < 1e-30

   1. Initial program 72.9%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 60.3

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.2

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

   7. Applied rewrites 39.2%

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

Alternative 10: 52.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := b \cdot \left(i \cdot t - c \cdot z\right)\\ \mathbf{if}\;b \leq -4.8 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-222}:\\ \;\;\;\;x \cdot \left(y \cdot z - a \cdot t\right)\\ \mathbf{elif}\;b \leq 10^{-30}:\\ \;\;\;\;j \cdot \left(a \cdot c - i \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* b (- (* i t) (* c z)))))
   (if (<= b -4.8e+59)
     t_1
     (if (<= b -3.6e-222)
       (* x (- (* y z) (* a t)))
       (if (<= b 1e-30) (* j (- (* a c) (* i y))) 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 * ((i * t) - (c * z));
	double tmp;
	if (b <= -4.8e+59) {
		tmp = t_1;
	} else if (b <= -3.6e-222) {
		tmp = x * ((y * z) - (a * t));
	} else if (b <= 1e-30) {
		tmp = j * ((a * c) - (i * y));
	} 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 * ((i * t) - (c * z))
    if (b <= (-4.8d+59)) then
        tmp = t_1
    else if (b <= (-3.6d-222)) then
        tmp = x * ((y * z) - (a * t))
    else if (b <= 1d-30) then
        tmp = j * ((a * c) - (i * y))
    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 * ((i * t) - (c * z));
	double tmp;
	if (b <= -4.8e+59) {
		tmp = t_1;
	} else if (b <= -3.6e-222) {
		tmp = x * ((y * z) - (a * t));
	} else if (b <= 1e-30) {
		tmp = j * ((a * c) - (i * y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = b * ((i * t) - (c * z))
	tmp = 0
	if b <= -4.8e+59:
		tmp = t_1
	elif b <= -3.6e-222:
		tmp = x * ((y * z) - (a * t))
	elif b <= 1e-30:
		tmp = j * ((a * c) - (i * y))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(b * Float64(Float64(i * t) - Float64(c * z)))
	tmp = 0.0
	if (b <= -4.8e+59)
		tmp = t_1;
	elseif (b <= -3.6e-222)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(a * t)));
	elseif (b <= 1e-30)
		tmp = Float64(j * Float64(Float64(a * c) - Float64(i * y)));
	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 * ((i * t) - (c * z));
	tmp = 0.0;
	if (b <= -4.8e+59)
		tmp = t_1;
	elseif (b <= -3.6e-222)
		tmp = x * ((y * z) - (a * t));
	elseif (b <= 1e-30)
		tmp = j * ((a * c) - (i * y));
	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[(N[(i * t), $MachinePrecision] - N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.8e+59], t$95$1, If[LessEqual[b, -3.6e-222], N[(x * N[(N[(y * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-30], N[(j * N[(N[(a * c), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.8000000000000004e59 or 1e-30 < b

   1. Initial program 72.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.3

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

   4. Applied rewrites 39.3%

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

   if -4.8000000000000004e59 < b < -3.59999999999999974e-222

   1. Initial program 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

   3. Applied rewrites 76.0%

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 38.8

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

   8. Applied rewrites 38.8%

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

   if -3.59999999999999974e-222 < b < 1e-30

   1. Initial program 72.9%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 60.3

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.2

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

   7. Applied rewrites 39.2%

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

Alternative 11: 52.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(a \cdot j - b \cdot z\right)\\ \mathbf{if}\;c \leq -1.65 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{+17}:\\ \;\;\;\;y \cdot \mathsf{fma}\left(-j, i, x \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (- (* a j) (* b z)))))
   (if (<= c -1.65e+123)
     t_1
     (if (<= c 8.5e+17) (* y (fma (- j) i (* x 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 = c * ((a * j) - (b * z));
	double tmp;
	if (c <= -1.65e+123) {
		tmp = t_1;
	} else if (c <= 8.5e+17) {
		tmp = y * fma(-j, i, (x * z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(Float64(a * j) - Float64(b * z)))
	tmp = 0.0
	if (c <= -1.65e+123)
		tmp = t_1;
	elseif (c <= 8.5e+17)
		tmp = Float64(y * fma(Float64(-j), i, Float64(x * 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[(c * N[(N[(a * j), $MachinePrecision] - N[(b * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.65e+123], t$95$1, If[LessEqual[c, 8.5e+17], N[(y * N[((-j) * i + N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.65000000000000001e123 or 8.5e17 < c

   1. Initial program 72.9%

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

   2. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 38.7

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

   4. Applied rewrites 38.7%

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

   if -1.65000000000000001e123 < c < 8.5e17

   1. Initial program 72.9%

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

   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. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.4

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

   4. Applied rewrites 39.4%

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

      1. lift-fma.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. distribute-lft-neg-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 39.8

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

   6. Applied rewrites 39.8%

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

Alternative 12: 51.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := c \cdot \left(a \cdot j - b \cdot z\right)\\ \mathbf{if}\;c \leq -1.65 \cdot 10^{+123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq 8.5 \cdot 10^{+17}:\\ \;\;\;\;\left(x \cdot z - j \cdot i\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* c (- (* a j) (* b z)))))
   (if (<= c -1.65e+123)
     t_1
     (if (<= c 8.5e+17) (* (- (* x z) (* j i)) y) 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 * j) - (b * z));
	double tmp;
	if (c <= -1.65e+123) {
		tmp = t_1;
	} else if (c <= 8.5e+17) {
		tmp = ((x * z) - (j * i)) * y;
	} 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 = c * ((a * j) - (b * z))
    if (c <= (-1.65d+123)) then
        tmp = t_1
    else if (c <= 8.5d+17) then
        tmp = ((x * z) - (j * i)) * y
    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 = c * ((a * j) - (b * z));
	double tmp;
	if (c <= -1.65e+123) {
		tmp = t_1;
	} else if (c <= 8.5e+17) {
		tmp = ((x * z) - (j * i)) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = c * ((a * j) - (b * z))
	tmp = 0
	if c <= -1.65e+123:
		tmp = t_1
	elif c <= 8.5e+17:
		tmp = ((x * z) - (j * i)) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(c * Float64(Float64(a * j) - Float64(b * z)))
	tmp = 0.0
	if (c <= -1.65e+123)
		tmp = t_1;
	elseif (c <= 8.5e+17)
		tmp = Float64(Float64(Float64(x * z) - Float64(j * i)) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = c * ((a * j) - (b * z));
	tmp = 0.0;
	if (c <= -1.65e+123)
		tmp = t_1;
	elseif (c <= 8.5e+17)
		tmp = ((x * z) - (j * i)) * y;
	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[(c * N[(N[(a * j), $MachinePrecision] - N[(b * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[c, -1.65e+123], t$95$1, If[LessEqual[c, 8.5e+17], N[(N[(N[(x * z), $MachinePrecision] - N[(j * i), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if c < -1.65000000000000001e123 or 8.5e17 < c

   1. Initial program 72.9%

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

   2. Taylor expanded in c around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 38.7

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

   4. Applied rewrites 38.7%

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

   if -1.65000000000000001e123 < c < 8.5e17

   1. Initial program 72.9%

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

   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. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.4

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

   4. Applied rewrites 39.4%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 39.4

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

      4. lift-fma.f64 N/A

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

      5. +-commutative N/A

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

      6. add-flip N/A

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

      7. lower--.f64 N/A

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

      8. distribute-lft-neg-out N/A

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

      9. metadata-eval N/A

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

      10. *-lft-identity 39.4

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 39.4

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

   6. Applied rewrites 39.4%

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

Alternative 13: 50.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := j \cdot \left(a \cdot c - i \cdot y\right)\\ \mathbf{if}\;j \leq -8.4 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;j \leq 4.6 \cdot 10^{-74}:\\ \;\;\;\;x \cdot \left(y \cdot z - a \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* j (- (* a c) (* i y)))))
   (if (<= j -8.4e-44) t_1 (if (<= j 4.6e-74) (* x (- (* y z) (* a 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 = j * ((a * c) - (i * y));
	double tmp;
	if (j <= -8.4e-44) {
		tmp = t_1;
	} else if (j <= 4.6e-74) {
		tmp = x * ((y * z) - (a * 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 = j * ((a * c) - (i * y))
    if (j <= (-8.4d-44)) then
        tmp = t_1
    else if (j <= 4.6d-74) then
        tmp = x * ((y * z) - (a * 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 = j * ((a * c) - (i * y));
	double tmp;
	if (j <= -8.4e-44) {
		tmp = t_1;
	} else if (j <= 4.6e-74) {
		tmp = x * ((y * z) - (a * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = j * ((a * c) - (i * y))
	tmp = 0
	if j <= -8.4e-44:
		tmp = t_1
	elif j <= 4.6e-74:
		tmp = x * ((y * z) - (a * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(j * Float64(Float64(a * c) - Float64(i * y)))
	tmp = 0.0
	if (j <= -8.4e-44)
		tmp = t_1;
	elseif (j <= 4.6e-74)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(a * 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 = j * ((a * c) - (i * y));
	tmp = 0.0;
	if (j <= -8.4e-44)
		tmp = t_1;
	elseif (j <= 4.6e-74)
		tmp = x * ((y * z) - (a * 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[(j * N[(N[(a * c), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[j, -8.4e-44], t$95$1, If[LessEqual[j, 4.6e-74], N[(x * N[(N[(y * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if j < -8.40000000000000005e-44 or 4.59999999999999961e-74 < j

   1. Initial program 72.9%

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

   2. Taylor expanded in b around 0

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

      1. lower-fma.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 60.3

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

   4. Applied rewrites 60.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.2

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

   7. Applied rewrites 39.2%

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

   if -8.40000000000000005e-44 < j < 4.59999999999999961e-74

   1. Initial program 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

   3. Applied rewrites 76.0%

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 38.8

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

   8. Applied rewrites 38.8%

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

Alternative 14: 41.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -3.6 \cdot 10^{+200}:\\ \;\;\;\;z \cdot \left(-1 \cdot \left(b \cdot c\right)\right)\\ \mathbf{elif}\;c \leq 2.8 \cdot 10^{+105}:\\ \;\;\;\;x \cdot \left(y \cdot z - a \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -3.6e+200)
   (* z (* -1.0 (* b c)))
   (if (<= c 2.8e+105) (* x (- (* y z) (* a t))) (* (* (- 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 (c <= -3.6e+200) {
		tmp = z * (-1.0 * (b * c));
	} else if (c <= 2.8e+105) {
		tmp = x * ((y * z) - (a * t));
	} else {
		tmp = (-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 (c <= (-3.6d+200)) then
        tmp = z * ((-1.0d0) * (b * c))
    else if (c <= 2.8d+105) then
        tmp = x * ((y * z) - (a * t))
    else
        tmp = (-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 (c <= -3.6e+200) {
		tmp = z * (-1.0 * (b * c));
	} else if (c <= 2.8e+105) {
		tmp = x * ((y * z) - (a * t));
	} else {
		tmp = (-c * z) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if c <= -3.6e+200:
		tmp = z * (-1.0 * (b * c))
	elif c <= 2.8e+105:
		tmp = x * ((y * z) - (a * t))
	else:
		tmp = (-c * z) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -3.6e+200)
		tmp = Float64(z * Float64(-1.0 * Float64(b * c)));
	elseif (c <= 2.8e+105)
		tmp = Float64(x * Float64(Float64(y * z) - Float64(a * t)));
	else
		tmp = Float64(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 (c <= -3.6e+200)
		tmp = z * (-1.0 * (b * c));
	elseif (c <= 2.8e+105)
		tmp = x * ((y * z) - (a * t));
	else
		tmp = (-c * z) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -3.6e+200], N[(z * N[(-1.0 * N[(b * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[c, 2.8e+105], N[(x * N[(N[(y * z), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -3.5999999999999998e200

   1. Initial program 72.9%

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

   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. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 38.9

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

   4. Applied rewrites 38.9%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 22.0

         \[\leadsto z \cdot \left(-1 \cdot \left(b \cdot c\right)\right) \]

   7. Applied rewrites 22.0%

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

   if -3.5999999999999998e200 < c < 2.8000000000000001e105

   1. Initial program 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

   3. Applied rewrites 76.0%

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 38.8

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

   8. Applied rewrites 38.8%

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

   if 2.8000000000000001e105 < c

   1. Initial program 72.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.3

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

   4. Applied rewrites 39.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 22.2

         \[\leadsto b \cdot \left(-1 \cdot \left(c \cdot z\right)\right) \]

   7. Applied rewrites 22.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 22.2

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

      4. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(c \cdot z\right)\right) \cdot b \]

      5. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(c \cdot z\right)\right) \cdot b \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot c\right) \cdot z\right) \cdot b \]

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 22.2

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

   9. Applied rewrites 22.2%

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

Alternative 15: 29.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot z\right) \cdot y\\ \mathbf{if}\;x \leq -4.2 \cdot 10^{+112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 3.45 \cdot 10^{-102}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{elif}\;x \leq 7 \cdot 10^{+30}:\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* x z) y)))
   (if (<= x -4.2e+112)
     t_1
     (if (<= x 3.45e-102)
       (* (* (- c) z) b)
       (if (<= x 7e+30) (* a (* c 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 = (x * z) * y;
	double tmp;
	if (x <= -4.2e+112) {
		tmp = t_1;
	} else if (x <= 3.45e-102) {
		tmp = (-c * z) * b;
	} else if (x <= 7e+30) {
		tmp = a * (c * 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 = (x * z) * y
    if (x <= (-4.2d+112)) then
        tmp = t_1
    else if (x <= 3.45d-102) then
        tmp = (-c * z) * b
    else if (x <= 7d+30) then
        tmp = a * (c * 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 = (x * z) * y;
	double tmp;
	if (x <= -4.2e+112) {
		tmp = t_1;
	} else if (x <= 3.45e-102) {
		tmp = (-c * z) * b;
	} else if (x <= 7e+30) {
		tmp = a * (c * j);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (x * z) * y
	tmp = 0
	if x <= -4.2e+112:
		tmp = t_1
	elif x <= 3.45e-102:
		tmp = (-c * z) * b
	elif x <= 7e+30:
		tmp = a * (c * j)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(x * z) * y)
	tmp = 0.0
	if (x <= -4.2e+112)
		tmp = t_1;
	elseif (x <= 3.45e-102)
		tmp = Float64(Float64(Float64(-c) * z) * b);
	elseif (x <= 7e+30)
		tmp = Float64(a * Float64(c * 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 = (x * z) * y;
	tmp = 0.0;
	if (x <= -4.2e+112)
		tmp = t_1;
	elseif (x <= 3.45e-102)
		tmp = (-c * z) * b;
	elseif (x <= 7e+30)
		tmp = a * (c * 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[(x * z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[x, -4.2e+112], t$95$1, If[LessEqual[x, 3.45e-102], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision], If[LessEqual[x, 7e+30], N[(a * N[(c * j), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.1999999999999998e112 or 7.00000000000000042e30 < x

   1. Initial program 72.9%

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

   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. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.4

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 22.1

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

   7. Applied rewrites 22.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 22.5

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

   9. Applied rewrites 22.5%

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

   if -4.1999999999999998e112 < x < 3.45e-102

   1. Initial program 72.9%

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

   2. Taylor expanded in b around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.3

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

   4. Applied rewrites 39.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 22.2

         \[\leadsto b \cdot \left(-1 \cdot \left(c \cdot z\right)\right) \]

   7. Applied rewrites 22.2%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 22.2

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

      4. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(c \cdot z\right)\right) \cdot b \]

      5. lift-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(c \cdot z\right)\right) \cdot b \]

      6. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot c\right) \cdot z\right) \cdot b \]

      7. mul-1-neg N/A

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

      8. lower-*.f64 N/A

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

      9. lower-neg.f64 22.2

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

   9. Applied rewrites 22.2%

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

   if 3.45e-102 < x < 7.00000000000000042e30

   1. Initial program 72.9%

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

   2. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 38.3

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

   4. Applied rewrites 38.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 21.9

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

   7. Applied rewrites 21.9%

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

Alternative 16: 29.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+98}:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+27}:\\ \;\;\;\;a \cdot \left(c \cdot j\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -1.3e+98)
   (* (* x z) y)
   (if (<= z 2.9e+27) (* a (* c j)) (* x (* y 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 <= -1.3e+98) {
		tmp = (x * z) * y;
	} else if (z <= 2.9e+27) {
		tmp = a * (c * j);
	} else {
		tmp = x * (y * 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 <= (-1.3d+98)) then
        tmp = (x * z) * y
    else if (z <= 2.9d+27) then
        tmp = a * (c * j)
    else
        tmp = x * (y * 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 <= -1.3e+98) {
		tmp = (x * z) * y;
	} else if (z <= 2.9e+27) {
		tmp = a * (c * j);
	} else {
		tmp = x * (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -1.3e+98:
		tmp = (x * z) * y
	elif z <= 2.9e+27:
		tmp = a * (c * j)
	else:
		tmp = x * (y * z)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -1.3e+98)
		tmp = Float64(Float64(x * z) * y);
	elseif (z <= 2.9e+27)
		tmp = Float64(a * Float64(c * j));
	else
		tmp = Float64(x * Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -1.3e+98)
		tmp = (x * z) * y;
	elseif (z <= 2.9e+27)
		tmp = a * (c * j);
	else
		tmp = x * (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -1.3e+98], N[(N[(x * z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 2.9e+27], N[(a * N[(c * j), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3e98

   1. Initial program 72.9%

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

   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. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.4

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 22.1

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

   7. Applied rewrites 22.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 22.5

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

   9. Applied rewrites 22.5%

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

   if -1.3e98 < z < 2.9000000000000001e27

   1. Initial program 72.9%

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

   2. Taylor expanded in 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. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 38.3

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

   4. Applied rewrites 38.3%

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

   5. Taylor expanded in x around 0

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

      1. lower-*.f64 21.9

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

   7. Applied rewrites 21.9%

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

   if 2.9000000000000001e27 < z

   1. Initial program 72.9%

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

   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. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.4

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 22.1

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

   7. Applied rewrites 22.1%

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

Alternative 17: 29.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+39}:\\ \;\;\;\;\left(x \cdot z\right) \cdot y\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-89}:\\ \;\;\;\;i \cdot \left(b \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= z -4.6e+39)
   (* (* x z) y)
   (if (<= z 1.15e-89) (* i (* b t)) (* x (* y 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 <= -4.6e+39) {
		tmp = (x * z) * y;
	} else if (z <= 1.15e-89) {
		tmp = i * (b * t);
	} else {
		tmp = x * (y * 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 <= (-4.6d+39)) then
        tmp = (x * z) * y
    else if (z <= 1.15d-89) then
        tmp = i * (b * t)
    else
        tmp = x * (y * 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 <= -4.6e+39) {
		tmp = (x * z) * y;
	} else if (z <= 1.15e-89) {
		tmp = i * (b * t);
	} else {
		tmp = x * (y * z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if z <= -4.6e+39:
		tmp = (x * z) * y
	elif z <= 1.15e-89:
		tmp = i * (b * t)
	else:
		tmp = x * (y * z)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (z <= -4.6e+39)
		tmp = Float64(Float64(x * z) * y);
	elseif (z <= 1.15e-89)
		tmp = Float64(i * Float64(b * t));
	else
		tmp = Float64(x * Float64(y * z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (z <= -4.6e+39)
		tmp = (x * z) * y;
	elseif (z <= 1.15e-89)
		tmp = i * (b * t);
	else
		tmp = x * (y * z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[z, -4.6e+39], N[(N[(x * z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 1.15e-89], N[(i * N[(b * t), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.60000000000000024e39

   1. Initial program 72.9%

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

   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. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.4

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 22.1

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

   7. Applied rewrites 22.1%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 22.5

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

   9. Applied rewrites 22.5%

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

   if -4.60000000000000024e39 < z < 1.15e-89

   1. Initial program 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

   3. Applied rewrites 76.0%

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.3

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

   8. Applied rewrites 39.3%

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

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 22.3

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

   11. Applied rewrites 22.3%

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

   if 1.15e-89 < z

   1. Initial program 72.9%

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

   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. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.4

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 22.1

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

   7. Applied rewrites 22.1%

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

Alternative 18: 29.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(y \cdot z\right)\\ \mathbf{if}\;y \leq -6 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-41}:\\ \;\;\;\;i \cdot \left(b \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* x (* y z))))
   (if (<= y -6e-72) t_1 (if (<= y 1.15e-41) (* 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 = x * (y * z);
	double tmp;
	if (y <= -6e-72) {
		tmp = t_1;
	} else if (y <= 1.15e-41) {
		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 = x * (y * z)
    if (y <= (-6d-72)) then
        tmp = t_1
    else if (y <= 1.15d-41) 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 = x * (y * z);
	double tmp;
	if (y <= -6e-72) {
		tmp = t_1;
	} else if (y <= 1.15e-41) {
		tmp = i * (b * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = x * (y * z)
	tmp = 0
	if y <= -6e-72:
		tmp = t_1
	elif y <= 1.15e-41:
		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(x * Float64(y * z))
	tmp = 0.0
	if (y <= -6e-72)
		tmp = t_1;
	elseif (y <= 1.15e-41)
		tmp = Float64(i * Float64(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 = x * (y * z);
	tmp = 0.0;
	if (y <= -6e-72)
		tmp = t_1;
	elseif (y <= 1.15e-41)
		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[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6e-72], t$95$1, If[LessEqual[y, 1.15e-41], N[(i * N[(b * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6e-72 or 1.15000000000000005e-41 < y

   1. Initial program 72.9%

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

   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. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.4

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

   4. Applied rewrites 39.4%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 22.1

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

   7. Applied rewrites 22.1%

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

   if -6e-72 < y < 1.15000000000000005e-41

   1. Initial program 72.9%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
   2. 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. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 74.7

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.7

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

      9. lift--.f64 N/A

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

      10. lift-*.f64 N/A

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

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

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

      12. +-commutative N/A

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

   3. Applied rewrites 76.0%

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in i around inf

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

      1. lower-*.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.3

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

   8. Applied rewrites 39.3%

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

   9. Taylor expanded in y around 0

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

       1. lower-*.f64 22.3

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

   11. Applied rewrites 22.3%

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

Alternative 19: 22.3% 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 72.9%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - t \cdot i\right)\right) + j \cdot \left(c \cdot a - y \cdot i\right) \]
2. 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. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 74.7

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 74.7

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

   9. lift--.f64 N/A

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

   10. lift-*.f64 N/A

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

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

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

   12. +-commutative N/A

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

3. Applied rewrites 76.0%

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

5. Applied rewrites 75.5%

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

6. Taylor expanded in i around inf

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

   1. lower-*.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-*.f64 39.3

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

8. Applied rewrites 39.3%

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

9. Taylor expanded in y around 0

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

    1. lower-*.f64 22.3

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

11. Applied rewrites 22.3%

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

Reproduce

herbie shell --seed 2025155 
(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)))))