Linear.Matrix:det33 from linear-1.19.1.3

Percentage Accurate: 73.0% → 81.4%

Time: 8.1s

Alternatives: 23

Speedup: 0.5×

• 57.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

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) - (i * a)))) + (j * ((c * t) - (i * y)));
}

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) - (i * a)))) + (j * ((c * t) - (i * y)))
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) - (i * a)))) + (j * ((c * t) - (i * y)));
}

Python

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

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(i * a)))) + Float64(j * Float64(Float64(c * t) - Float64(i * y))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
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[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 73.0% accurate, 1.0× speedup

Math

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

FPCore

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

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) - (i * a)))) + (j * ((c * t) - (i * y)));
}

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) - (i * a)))) + (j * ((c * t) - (i * y)))
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) - (i * a)))) + (j * ((c * t) - (i * y)));
}

Python

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

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(i * a)))) + Float64(j * Float64(Float64(c * t) - Float64(i * y))))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + (j * ((c * t) - (i * y)));
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[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 81.4% accurate, 0.5× speedup

Math

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

FPCore

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

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

Julia

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

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(j * N[(N[(c * t), $MachinePrecision] - N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], Infinity], N[(t$95$1 * b + N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[(N[((-i) * y + N[(c * t), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * b), $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 i a)))) (*.f64 j (-.f64 (*.f64 c t) (*.f64 i y)))) < +inf.0

   1. Initial program 88.8%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 88.8%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 53.4

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

   5. Applied rewrites 53.4%

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

Alternative 2: 66.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq -1.12 \cdot 10^{+94} \lor \neg \left(j \leq 0.00023\right):\\ \;\;\;\;\mathsf{fma}\left(-i, j \cdot y, t \cdot \mathsf{fma}\left(-a, x, c \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(-b\right) \cdot \mathsf{fma}\left(-a, i, c \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= j -1.12e+94) (not (<= j 0.00023)))
   (fma (- i) (* j y) (* t (fma (- a) x (* c j))))
   (fma (fma (- a) t (* z y)) x (* (- b) (fma (- a) i (* c z))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((j <= -1.12e+94) || !(j <= 0.00023)) {
		tmp = fma(-i, (j * y), (t * fma(-a, x, (c * j))));
	} else {
		tmp = fma(fma(-a, t, (z * y)), x, (-b * fma(-a, i, (c * z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((j <= -1.12e+94) || !(j <= 0.00023))
		tmp = fma(Float64(-i), Float64(j * y), Float64(t * fma(Float64(-a), x, Float64(c * j))));
	else
		tmp = fma(fma(Float64(-a), t, Float64(z * y)), x, Float64(Float64(-b) * fma(Float64(-a), i, Float64(c * z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[j, -1.12e+94], N[Not[LessEqual[j, 0.00023]], $MachinePrecision]], N[((-i) * N[(j * y), $MachinePrecision] + N[(t * N[((-a) * x + N[(c * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x + N[((-b) * N[((-a) * i + N[(c * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if j < -1.11999999999999996e94 or 2.3000000000000001e-4 < j

   1. Initial program 67.7%

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

   3. Taylor expanded in c around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

   5. Applied rewrites 59.0%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(-i, j \cdot y, t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lower-*.f64 69.6

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

   8. Applied rewrites 69.6%

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

   if -1.11999999999999996e94 < j < 2.3000000000000001e-4

   1. Initial program 70.9%

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

   3. Taylor expanded in j around 0

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

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

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

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

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

      9. +-commutative N/A

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

      10. associate-*r* N/A

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

      11. mul-1-neg N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-neg.f64 N/A

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

      14. *-commutative N/A

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

      15. lower-*.f64 N/A

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

   5. Applied rewrites 77.4%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;j \leq -1.12 \cdot 10^{+94} \lor \neg \left(j \leq 0.00023\right):\\ \;\;\;\;\mathsf{fma}\left(-i, j \cdot y, t \cdot \mathsf{fma}\left(-a, x, c \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(-a, t, z \cdot y\right), x, \left(-b\right) \cdot \mathsf{fma}\left(-a, i, c \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 63.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{if}\;y \leq -2 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-179}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(-b, z, j \cdot t\right), x \cdot \mathsf{fma}\left(-a, t, y \cdot z\right)\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right), b, \left(-a\right) \cdot \left(t \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (fma (- i) j (* z x)) y)))
   (if (<= y -2e+92)
     t_1
     (if (<= y 2.1e-179)
       (fma c (fma (- b) z (* j t)) (* x (fma (- a) t (* y z))))
       (if (<= y 1.15e+62)
         (fma (fma i a (* (- c) z)) b (* (- a) (* t x)))
         t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = fma(-i, j, (z * x)) * y;
	double tmp;
	if (y <= -2e+92) {
		tmp = t_1;
	} else if (y <= 2.1e-179) {
		tmp = fma(c, fma(-b, z, (j * t)), (x * fma(-a, t, (y * z))));
	} else if (y <= 1.15e+62) {
		tmp = fma(fma(i, a, (-c * z)), b, (-a * (t * x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(fma(Float64(-i), j, Float64(z * x)) * y)
	tmp = 0.0
	if (y <= -2e+92)
		tmp = t_1;
	elseif (y <= 2.1e-179)
		tmp = fma(c, fma(Float64(-b), z, Float64(j * t)), Float64(x * fma(Float64(-a), t, Float64(y * z))));
	elseif (y <= 1.15e+62)
		tmp = fma(fma(i, a, Float64(Float64(-c) * z)), b, Float64(Float64(-a) * Float64(t * x)));
	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[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2e+92], t$95$1, If[LessEqual[y, 2.1e-179], N[(c * N[((-b) * z + N[(j * t), $MachinePrecision]), $MachinePrecision] + N[(x * N[((-a) * t + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.15e+62], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b + N[((-a) * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.0000000000000001e92 or 1.14999999999999992e62 < y

   1. Initial program 61.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 75.1

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

   5. Applied rewrites 75.1%

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

   if -2.0000000000000001e92 < y < 2.0999999999999999e-179

   1. Initial program 71.2%

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

   3. Taylor expanded in c around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

   5. Applied rewrites 78.2%

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

   6. Taylor expanded in i around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-*.f64 70.4

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

   8. Applied rewrites 70.4%

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

   if 2.0999999999999999e-179 < y < 1.14999999999999992e62

   1. Initial program 79.7%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 83.3%

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

   6. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 78.1

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

   8. Applied rewrites 78.1%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;y \leq 2.1 \cdot 10^{-179}:\\ \;\;\;\;\mathsf{fma}\left(c, \mathsf{fma}\left(-b, z, j \cdot t\right), x \cdot \mathsf{fma}\left(-a, t, y \cdot z\right)\right)\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{+62}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right), b, \left(-a\right) \cdot \left(t \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 59.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.1 \cdot 10^{-16} \lor \neg \left(y \leq 1.15 \cdot 10^{+62}\right):\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right), b, \left(-a\right) \cdot \left(t \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= y -7.1e-16) (not (<= y 1.15e+62)))
   (* (fma (- i) j (* z x)) y)
   (fma (fma i a (* (- c) z)) b (* (- a) (* t x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((y <= -7.1e-16) || !(y <= 1.15e+62)) {
		tmp = fma(-i, j, (z * x)) * y;
	} else {
		tmp = fma(fma(i, a, (-c * z)), b, (-a * (t * x)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((y <= -7.1e-16) || !(y <= 1.15e+62))
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	else
		tmp = fma(fma(i, a, Float64(Float64(-c) * z)), b, Float64(Float64(-a) * Float64(t * x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[y, -7.1e-16], N[Not[LessEqual[y, 1.15e+62]], $MachinePrecision]], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b + N[((-a) * N[(t * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.1e-16 or 1.14999999999999992e62 < y

   1. Initial program 60.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.0

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

   5. Applied rewrites 72.0%

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

   if -7.1e-16 < y < 1.14999999999999992e62

   1. Initial program 76.0%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 79.3%

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

   6. Taylor expanded in a around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 70.6

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

   8. Applied rewrites 70.6%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right), b, -a \cdot \left(t \cdot x\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.1 \cdot 10^{-16} \lor \neg \left(y \leq 1.15 \cdot 10^{+62}\right):\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right), b, \left(-a\right) \cdot \left(t \cdot x\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 60.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+58} \lor \neg \left(t \leq 8 \cdot 10^{+64}\right):\\ \;\;\;\;\mathsf{fma}\left(-i, j \cdot y, t \cdot \mathsf{fma}\left(-a, x, c \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right), b, x \cdot \left(y \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= t -2.1e+58) (not (<= t 8e+64)))
   (fma (- i) (* j y) (* t (fma (- a) x (* c j))))
   (fma (fma i a (* (- c) z)) b (* 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 ((t <= -2.1e+58) || !(t <= 8e+64)) {
		tmp = fma(-i, (j * y), (t * fma(-a, x, (c * j))));
	} else {
		tmp = fma(fma(i, a, (-c * z)), b, (x * (y * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((t <= -2.1e+58) || !(t <= 8e+64))
		tmp = fma(Float64(-i), Float64(j * y), Float64(t * fma(Float64(-a), x, Float64(c * j))));
	else
		tmp = fma(fma(i, a, Float64(Float64(-c) * z)), b, Float64(x * Float64(y * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[t, -2.1e+58], N[Not[LessEqual[t, 8e+64]], $MachinePrecision]], N[((-i) * N[(j * y), $MachinePrecision] + N[(t * N[((-a) * x + N[(c * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.10000000000000012e58 or 8.00000000000000017e64 < t

   1. Initial program 58.3%

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

   3. Taylor expanded in c around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(-i, j \cdot y, t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)\right) \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lower-*.f64 74.2

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

   8. Applied rewrites 74.2%

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

   if -2.10000000000000012e58 < t < 8.00000000000000017e64

   1. Initial program 76.9%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 78.8%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 66.0

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

   8. Applied rewrites 66.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right), b, x \cdot \left(y \cdot z\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+58} \lor \neg \left(t \leq 8 \cdot 10^{+64}\right):\\ \;\;\;\;\mathsf{fma}\left(-i, j \cdot y, t \cdot \mathsf{fma}\left(-a, x, c \cdot j\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right), b, x \cdot \left(y \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 58.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{elif}\;t \leq 8 \cdot 10^{+64}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right), b, x \cdot \left(y \cdot z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot \mathsf{fma}\left(a, \frac{x}{c}, -j\right)\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= t -1.85e+60)
   (* (fma (- a) x (* j c)) t)
   (if (<= t 8e+64)
     (fma (fma i a (* (- c) z)) b (* x (* y z)))
     (* (* (- c) (fma a (/ x c) (- j))) t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (t <= -1.85e+60) {
		tmp = fma(-a, x, (j * c)) * t;
	} else if (t <= 8e+64) {
		tmp = fma(fma(i, a, (-c * z)), b, (x * (y * z)));
	} else {
		tmp = (-c * fma(a, (x / c), -j)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (t <= -1.85e+60)
		tmp = Float64(fma(Float64(-a), x, Float64(j * c)) * t);
	elseif (t <= 8e+64)
		tmp = fma(fma(i, a, Float64(Float64(-c) * z)), b, Float64(x * Float64(y * z)));
	else
		tmp = Float64(Float64(Float64(-c) * fma(a, Float64(x / c), Float64(-j))) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[t, -1.85e+60], N[(N[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[t, 8e+64], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-c) * N[(a * N[(x / c), $MachinePrecision] + (-j)), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.84999999999999994e60

   1. Initial program 54.8%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 70.6

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

   5. Applied rewrites 70.6%

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

   if -1.84999999999999994e60 < t < 8.00000000000000017e64

   1. Initial program 76.9%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 78.8%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 66.0

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

   8. Applied rewrites 66.0%

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

   if 8.00000000000000017e64 < t

   1. Initial program 61.1%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 59.6

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

   5. Applied rewrites 59.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(-1 \cdot \left(a \cdot x\right)\right) \cdot t \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 40.3

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

   8. Applied rewrites 40.3%

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

   9. Taylor expanded in c around -inf

      \[\leadsto \left(-1 \cdot \left(c \cdot \left(-1 \cdot j + \frac{a \cdot x}{c}\right)\right)\right) \cdot t \]
   10. Step-by-step derivation

       1. associate-*r* N/A

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

       2. lower-*.f64 N/A

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

       3. mul-1-neg N/A

          \[\leadsto \left(\left(\mathsf{neg}\left(c\right)\right) \cdot \left(-1 \cdot j + \frac{a \cdot x}{c}\right)\right) \cdot t \]

       4. lower-neg.f64 N/A

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

       5. +-commutative N/A

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

       6. associate-/l* N/A

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

       7. lower-fma.f64 N/A

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

       8. lower-/.f64 N/A

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

       9. mul-1-neg N/A

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

       10. lower-neg.f64 59.7

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

   11. Applied rewrites 59.7%

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

Alternative 7: 58.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+60} \lor \neg \left(t \leq 8 \cdot 10^{+64}\right):\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right), b, x \cdot \left(y \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= t -1.85e+60) (not (<= t 8e+64)))
   (* (fma (- a) x (* j c)) t)
   (fma (fma i a (* (- c) z)) b (* 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 ((t <= -1.85e+60) || !(t <= 8e+64)) {
		tmp = fma(-a, x, (j * c)) * t;
	} else {
		tmp = fma(fma(i, a, (-c * z)), b, (x * (y * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((t <= -1.85e+60) || !(t <= 8e+64))
		tmp = Float64(fma(Float64(-a), x, Float64(j * c)) * t);
	else
		tmp = fma(fma(i, a, Float64(Float64(-c) * z)), b, Float64(x * Float64(y * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[t, -1.85e+60], N[Not[LessEqual[t, 8e+64]], $MachinePrecision]], N[(N[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b + N[(x * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.84999999999999994e60 or 8.00000000000000017e64 < t

   1. Initial program 58.3%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 64.4

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

   5. Applied rewrites 64.4%

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

   if -1.84999999999999994e60 < t < 8.00000000000000017e64

   1. Initial program 76.9%

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

   3. Taylor expanded in b around 0

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

   5. Applied rewrites 78.8%

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

   6. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 66.0

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

   8. Applied rewrites 66.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right), b, x \cdot \left(y \cdot z\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 65.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.85 \cdot 10^{+60} \lor \neg \left(t \leq 8 \cdot 10^{+64}\right):\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right), b, x \cdot \left(y \cdot z\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -5.5 \cdot 10^{+94}:\\ \;\;\;\;\mathsf{fma}\left(j, t, \left(-b\right) \cdot z\right) \cdot c\\ \mathbf{elif}\;c \leq -2.3 \cdot 10^{-242}:\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{elif}\;c \leq 2.7 \cdot 10^{-24}:\\ \;\;\;\;\left(-a\right) \cdot \mathsf{fma}\left(t, x, \left(-b\right) \cdot i\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-b, z, j \cdot t\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -5.5e+94)
   (* (fma j t (* (- b) z)) c)
   (if (<= c -2.3e-242)
     (* (fma (- i) j (* z x)) y)
     (if (<= c 2.7e-24)
       (* (- a) (fma t x (* (- b) i)))
       (* (fma (- b) z (* j t)) c)))))

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 <= -5.5e+94) {
		tmp = fma(j, t, (-b * z)) * c;
	} else if (c <= -2.3e-242) {
		tmp = fma(-i, j, (z * x)) * y;
	} else if (c <= 2.7e-24) {
		tmp = -a * fma(t, x, (-b * i));
	} else {
		tmp = fma(-b, z, (j * t)) * c;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -5.5e+94)
		tmp = Float64(fma(j, t, Float64(Float64(-b) * z)) * c);
	elseif (c <= -2.3e-242)
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	elseif (c <= 2.7e-24)
		tmp = Float64(Float64(-a) * fma(t, x, Float64(Float64(-b) * i)));
	else
		tmp = Float64(fma(Float64(-b), z, Float64(j * t)) * c);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -5.5e+94], N[(N[(j * t + N[((-b) * z), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, -2.3e-242], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[c, 2.7e-24], N[((-a) * N[(t * x + N[((-b) * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-b) * z + N[(j * t), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if c < -5.4999999999999997e94

   1. Initial program 51.8%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 82.4

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

   5. Applied rewrites 82.4%

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

   if -5.4999999999999997e94 < c < -2.29999999999999985e-242

   1. Initial program 76.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 57.7

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

   5. Applied rewrites 57.7%

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

   if -2.29999999999999985e-242 < c < 2.70000000000000007e-24

   1. Initial program 75.3%

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

   3. Taylor expanded in a around -inf

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-neg.f64 59.0

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

   5. Applied rewrites 59.0%

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

   if 2.70000000000000007e-24 < c

   1. Initial program 65.8%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 61.7

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

   5. Applied rewrites 61.7%

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

   6. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

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

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

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

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-*.f64 64.2

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

   8. Applied rewrites 64.2%

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

Alternative 9: 43.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+93}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;y \leq -0.000155 \lor \neg \left(y \leq 1.05 \cdot 10^{+49}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -6e+93)
   (* (* (- i) j) y)
   (if (or (<= y -0.000155) (not (<= y 1.05e+49)))
     (* (fma y x (* (- b) c)) z)
     (* (fma i a (* (- 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 (y <= -6e+93) {
		tmp = (-i * j) * y;
	} else if ((y <= -0.000155) || !(y <= 1.05e+49)) {
		tmp = fma(y, x, (-b * c)) * z;
	} else {
		tmp = fma(i, a, (-c * z)) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -6e+93)
		tmp = Float64(Float64(Float64(-i) * j) * y);
	elseif ((y <= -0.000155) || !(y <= 1.05e+49))
		tmp = Float64(fma(y, x, Float64(Float64(-b) * c)) * z);
	else
		tmp = Float64(fma(i, a, Float64(Float64(-c) * z)) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -6e+93], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision], If[Or[LessEqual[y, -0.000155], N[Not[LessEqual[y, 1.05e+49]], $MachinePrecision]], N[(N[(y * x + N[((-b) * c), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.99999999999999957e93

   1. Initial program 62.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.4

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(i \cdot j\right)\right) \cdot y \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 56.1

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

   8. Applied rewrites 56.1%

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

   if -5.99999999999999957e93 < y < -1.55e-4 or 1.05000000000000005e49 < y

   1. Initial program 60.7%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 61.5

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

   5. Applied rewrites 61.5%

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

   if -1.55e-4 < y < 1.05000000000000005e49

   1. Initial program 75.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 55.2

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

   5. Applied rewrites 55.2%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+93}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;y \leq -0.000155 \lor \neg \left(y \leq 1.05 \cdot 10^{+49}\right):\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 41.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+95}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(j, t, \left(-b\right) \cdot z\right) \cdot c\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -1.55e+95)
   (* (* (- i) j) y)
   (if (<= y 2.8e-299)
     (* (fma j t (* (- b) z)) c)
     (if (<= y 1.25e+74) (* (fma i a (* (- c) z)) b) (* (* z y) x)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -1.55e+95) {
		tmp = (-i * j) * y;
	} else if (y <= 2.8e-299) {
		tmp = fma(j, t, (-b * z)) * c;
	} else if (y <= 1.25e+74) {
		tmp = fma(i, a, (-c * z)) * b;
	} else {
		tmp = (z * y) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -1.55e+95)
		tmp = Float64(Float64(Float64(-i) * j) * y);
	elseif (y <= 2.8e-299)
		tmp = Float64(fma(j, t, Float64(Float64(-b) * z)) * c);
	elseif (y <= 1.25e+74)
		tmp = Float64(fma(i, a, Float64(Float64(-c) * z)) * b);
	else
		tmp = Float64(Float64(z * y) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -1.55e+95], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 2.8e-299], N[(N[(j * t + N[((-b) * z), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[y, 1.25e+74], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.5500000000000001e95

   1. Initial program 62.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 74.4

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

   5. Applied rewrites 74.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(i \cdot j\right)\right) \cdot y \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 56.1

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

   8. Applied rewrites 56.1%

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

   if -1.5500000000000001e95 < y < 2.8000000000000001e-299

   1. Initial program 68.7%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 54.8

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

   5. Applied rewrites 54.8%

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

   if 2.8000000000000001e-299 < y < 1.24999999999999991e74

   1. Initial program 79.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 55.7

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

   5. Applied rewrites 55.7%

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

   if 1.24999999999999991e74 < y

   1. Initial program 58.4%

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

   3. Taylor expanded in c around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in i around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-*.f64 48.1

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

   8. Applied rewrites 48.1%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 47.6

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

   11. Applied rewrites 47.6%

       \[\leadsto \left(z \cdot y\right) \cdot x \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+95}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-299}:\\ \;\;\;\;\mathsf{fma}\left(j, t, \left(-b\right) \cdot z\right) \cdot c\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{+74}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 29.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+20}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-49}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-239}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+74}:\\ \;\;\;\;\left(a \cdot i\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= y -5e+20)
   (* (* z x) y)
   (if (<= y -5.2e-49)
     (* (* b a) i)
     (if (<= y 1.8e-239)
       (* (- a) (* t x))
       (if (<= y 1.05e+74) (* (* a i) b) (* (* z y) x))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (y <= -5e+20) {
		tmp = (z * x) * y;
	} else if (y <= -5.2e-49) {
		tmp = (b * a) * i;
	} else if (y <= 1.8e-239) {
		tmp = -a * (t * x);
	} else if (y <= 1.05e+74) {
		tmp = (a * i) * b;
	} else {
		tmp = (z * y) * x;
	}
	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 (y <= (-5d+20)) then
        tmp = (z * x) * y
    else if (y <= (-5.2d-49)) then
        tmp = (b * a) * i
    else if (y <= 1.8d-239) then
        tmp = -a * (t * x)
    else if (y <= 1.05d+74) then
        tmp = (a * i) * b
    else
        tmp = (z * y) * x
    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 (y <= -5e+20) {
		tmp = (z * x) * y;
	} else if (y <= -5.2e-49) {
		tmp = (b * a) * i;
	} else if (y <= 1.8e-239) {
		tmp = -a * (t * x);
	} else if (y <= 1.05e+74) {
		tmp = (a * i) * b;
	} else {
		tmp = (z * y) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if y <= -5e+20:
		tmp = (z * x) * y
	elif y <= -5.2e-49:
		tmp = (b * a) * i
	elif y <= 1.8e-239:
		tmp = -a * (t * x)
	elif y <= 1.05e+74:
		tmp = (a * i) * b
	else:
		tmp = (z * y) * x
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (y <= -5e+20)
		tmp = Float64(Float64(z * x) * y);
	elseif (y <= -5.2e-49)
		tmp = Float64(Float64(b * a) * i);
	elseif (y <= 1.8e-239)
		tmp = Float64(Float64(-a) * Float64(t * x));
	elseif (y <= 1.05e+74)
		tmp = Float64(Float64(a * i) * b);
	else
		tmp = Float64(Float64(z * y) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if (y <= -5e+20)
		tmp = (z * x) * y;
	elseif (y <= -5.2e-49)
		tmp = (b * a) * i;
	elseif (y <= 1.8e-239)
		tmp = -a * (t * x);
	elseif (y <= 1.05e+74)
		tmp = (a * i) * b;
	else
		tmp = (z * y) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[y, -5e+20], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, -5.2e-49], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision], If[LessEqual[y, 1.8e-239], N[((-a) * N[(t * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.05e+74], N[(N[(a * i), $MachinePrecision] * b), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -5e20

   1. Initial program 63.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 73.7

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

   5. Applied rewrites 73.7%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lift-*.f64 36.0

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

   8. Applied rewrites 36.0%

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

   if -5e20 < y < -5.1999999999999999e-49

   1. Initial program 57.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 58.3

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 51.9

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

   8. Applied rewrites 51.9%

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

      1. lift-*.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 51.9

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

   10. Applied rewrites 51.9%

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

   if -5.1999999999999999e-49 < y < 1.8000000000000001e-239

   1. Initial program 74.2%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 53.1

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

   5. Applied rewrites 53.1%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 39.5

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

   8. Applied rewrites 39.5%

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

   if 1.8000000000000001e-239 < y < 1.0499999999999999e74

   1. Initial program 78.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 58.9

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

   5. Applied rewrites 58.9%

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

   6. Taylor expanded in z around 0

      \[\leadsto \left(a \cdot i\right) \cdot b \]
   7. Step-by-step derivation

      1. lower-*.f64 31.4

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

   8. Applied rewrites 31.4%

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

   if 1.0499999999999999e74 < y

   1. Initial program 58.4%

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

   3. Taylor expanded in c around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in i around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lift-neg.f64 N/A

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

      12. lower-*.f64 48.1

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

   8. Applied rewrites 48.1%

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

   9. Taylor expanded in y around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 47.6

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

   11. Applied rewrites 47.6%

       \[\leadsto \left(z \cdot y\right) \cdot x \]
3. Recombined 5 regimes into one program.

4. Final simplification 38.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+20}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-49}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-239}:\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+74}:\\ \;\;\;\;\left(a \cdot i\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 52.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+21} \lor \neg \left(y \leq 9.8 \cdot 10^{+43}\right):\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= y -4.8e+21) (not (<= y 9.8e+43)))
   (* (fma (- i) j (* z x)) y)
   (* (fma i a (* (- 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 ((y <= -4.8e+21) || !(y <= 9.8e+43)) {
		tmp = fma(-i, j, (z * x)) * y;
	} else {
		tmp = fma(i, a, (-c * z)) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((y <= -4.8e+21) || !(y <= 9.8e+43))
		tmp = Float64(fma(Float64(-i), j, Float64(z * x)) * y);
	else
		tmp = Float64(fma(i, a, Float64(Float64(-c) * z)) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[y, -4.8e+21], N[Not[LessEqual[y, 9.8e+43]], $MachinePrecision]], N[(N[((-i) * j + N[(z * x), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8e21 or 9.7999999999999999e43 < y

   1. Initial program 62.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 72.2

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

   5. Applied rewrites 72.2%

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

   if -4.8e21 < y < 9.7999999999999999e43

   1. Initial program 74.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 54.8

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

   5. Applied rewrites 54.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{+21} \lor \neg \left(y \leq 9.8 \cdot 10^{+43}\right):\\ \;\;\;\;\mathsf{fma}\left(-i, j, z \cdot x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 51.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+59} \lor \neg \left(t \leq 75000000000000\right):\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= t -6e+59) (not (<= t 75000000000000.0)))
   (* (fma (- a) x (* j c)) t)
   (* (fma y x (* (- b) c)) z)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((t <= -6e+59) || !(t <= 75000000000000.0)) {
		tmp = fma(-a, x, (j * c)) * t;
	} else {
		tmp = fma(y, x, (-b * c)) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((t <= -6e+59) || !(t <= 75000000000000.0))
		tmp = Float64(fma(Float64(-a), x, Float64(j * c)) * t);
	else
		tmp = Float64(fma(y, x, Float64(Float64(-b) * c)) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[t, -6e+59], N[Not[LessEqual[t, 75000000000000.0]], $MachinePrecision]], N[(N[((-a) * x + N[(j * c), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[(y * x + N[((-b) * c), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -6.0000000000000001e59 or 7.5e13 < t

   1. Initial program 59.0%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 62.8

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

   5. Applied rewrites 62.8%

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

   if -6.0000000000000001e59 < t < 7.5e13

   1. Initial program 77.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower-neg.f64 56.3

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

   5. Applied rewrites 56.3%

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

4. Final simplification 59.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{+59} \lor \neg \left(t \leq 75000000000000\right):\\ \;\;\;\;\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, \left(-b\right) \cdot c\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\left(-c\right) \cdot z\right) \cdot b\\ \mathbf{if}\;c \leq -5.8 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -4.2 \cdot 10^{-246}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \mathbf{elif}\;c \leq 3 \cdot 10^{-24}:\\ \;\;\;\;\left(a \cdot i\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* (- c) z) b)))
   (if (<= c -5.8e+34)
     t_1
     (if (<= c -4.2e-246)
       (* (* (- i) j) y)
       (if (<= c 3e-24) (* (* a i) b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-c * z) * b;
	double tmp;
	if (c <= -5.8e+34) {
		tmp = t_1;
	} else if (c <= -4.2e-246) {
		tmp = (-i * j) * y;
	} else if (c <= 3e-24) {
		tmp = (a * i) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (-c * z) * b
    if (c <= (-5.8d+34)) then
        tmp = t_1
    else if (c <= (-4.2d-246)) then
        tmp = (-i * j) * y
    else if (c <= 3d-24) then
        tmp = (a * i) * b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (-c * z) * b;
	double tmp;
	if (c <= -5.8e+34) {
		tmp = t_1;
	} else if (c <= -4.2e-246) {
		tmp = (-i * j) * y;
	} else if (c <= 3e-24) {
		tmp = (a * i) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (-c * z) * b
	tmp = 0
	if c <= -5.8e+34:
		tmp = t_1
	elif c <= -4.2e-246:
		tmp = (-i * j) * y
	elif c <= 3e-24:
		tmp = (a * i) * b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(-c) * z) * b)
	tmp = 0.0
	if (c <= -5.8e+34)
		tmp = t_1;
	elseif (c <= -4.2e-246)
		tmp = Float64(Float64(Float64(-i) * j) * y);
	elseif (c <= 3e-24)
		tmp = Float64(Float64(a * i) * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (-c * z) * b;
	tmp = 0.0;
	if (c <= -5.8e+34)
		tmp = t_1;
	elseif (c <= -4.2e-246)
		tmp = (-i * j) * y;
	elseif (c <= 3e-24)
		tmp = (a * i) * b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision]}, If[LessEqual[c, -5.8e+34], t$95$1, If[LessEqual[c, -4.2e-246], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[c, 3e-24], N[(N[(a * i), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if c < -5.8000000000000003e34 or 2.99999999999999995e-24 < c

   1. Initial program 60.5%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 60.7

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

   5. Applied rewrites 60.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot \left(c \cdot z\right)\right) \cdot b \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 51.2

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

   8. Applied rewrites 51.2%

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

   if -5.8000000000000003e34 < c < -4.19999999999999989e-246

   1. Initial program 77.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 56.8

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(i \cdot j\right)\right) \cdot y \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 36.6

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

   8. Applied rewrites 36.6%

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

   if -4.19999999999999989e-246 < c < 2.99999999999999995e-24

   1. Initial program 75.3%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 42.0

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in z around 0

      \[\leadsto \left(a \cdot i\right) \cdot b \]
   7. Step-by-step derivation

      1. lower-*.f64 37.1

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

   8. Applied rewrites 37.1%

      \[\leadsto \left(a \cdot i\right) \cdot b \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 47.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq -2.05 \cdot 10^{+124}:\\ \;\;\;\;\mathsf{fma}\left(j, t, \left(-b\right) \cdot z\right) \cdot c\\ \mathbf{elif}\;c \leq 8 \cdot 10^{-122}:\\ \;\;\;\;\mathsf{fma}\left(-a, t, z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= c -2.05e+124)
   (* (fma j t (* (- b) z)) c)
   (if (<= c 8e-122) (* (fma (- a) t (* z y)) x) (* (fma i a (* (- 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 <= -2.05e+124) {
		tmp = fma(j, t, (-b * z)) * c;
	} else if (c <= 8e-122) {
		tmp = fma(-a, t, (z * y)) * x;
	} else {
		tmp = fma(i, a, (-c * z)) * b;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (c <= -2.05e+124)
		tmp = Float64(fma(j, t, Float64(Float64(-b) * z)) * c);
	elseif (c <= 8e-122)
		tmp = Float64(fma(Float64(-a), t, Float64(z * y)) * x);
	else
		tmp = Float64(fma(i, a, Float64(Float64(-c) * z)) * b);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[c, -2.05e+124], N[(N[(j * t + N[((-b) * z), $MachinePrecision]), $MachinePrecision] * c), $MachinePrecision], If[LessEqual[c, 8e-122], N[(N[((-a) * t + N[(z * y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if c < -2.05000000000000001e124

   1. Initial program 47.2%

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

   3. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-neg.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if -2.05000000000000001e124 < c < 8.00000000000000047e-122

   1. Initial program 77.4%

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

   3. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

         \[\leadsto \left(y \cdot z + \left(-1 \cdot a\right) \cdot t\right) \cdot x \]

      7. associate-*r* N/A

         \[\leadsto \left(y \cdot z + -1 \cdot \left(a \cdot t\right)\right) \cdot x \]

      8. +-commutative N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot t\right) + y \cdot z\right) \cdot x \]

      9. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot a\right) \cdot t + y \cdot z\right) \cdot x \]

      10. mul-1-neg N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 44.9

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

   5. Applied rewrites 44.9%

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

   if 8.00000000000000047e-122 < c

   1. Initial program 67.0%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 58.3

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

   5. Applied rewrites 58.3%

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

Alternative 16: 27.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(c \cdot j\right) \cdot t\\ \mathbf{if}\;c \leq -6.8 \cdot 10^{+169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;c \leq -3.1 \cdot 10^{-242}:\\ \;\;\;\;\left(z \cdot x\right) \cdot y\\ \mathbf{elif}\;c \leq 8 \cdot 10^{+208}:\\ \;\;\;\;\left(a \cdot i\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1 (* (* c j) t)))
   (if (<= c -6.8e+169)
     t_1
     (if (<= c -3.1e-242)
       (* (* z x) y)
       (if (<= c 8e+208) (* (* a i) b) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (c * j) * t;
	double tmp;
	if (c <= -6.8e+169) {
		tmp = t_1;
	} else if (c <= -3.1e-242) {
		tmp = (z * x) * y;
	} else if (c <= 8e+208) {
		tmp = (a * i) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (c * j) * t
    if (c <= (-6.8d+169)) then
        tmp = t_1
    else if (c <= (-3.1d-242)) then
        tmp = (z * x) * y
    else if (c <= 8d+208) then
        tmp = (a * i) * b
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = (c * j) * t;
	double tmp;
	if (c <= -6.8e+169) {
		tmp = t_1;
	} else if (c <= -3.1e-242) {
		tmp = (z * x) * y;
	} else if (c <= 8e+208) {
		tmp = (a * i) * b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = (c * j) * t
	tmp = 0
	if c <= -6.8e+169:
		tmp = t_1
	elif c <= -3.1e-242:
		tmp = (z * x) * y
	elif c <= 8e+208:
		tmp = (a * i) * b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(c * j) * t)
	tmp = 0.0
	if (c <= -6.8e+169)
		tmp = t_1;
	elseif (c <= -3.1e-242)
		tmp = Float64(Float64(z * x) * y);
	elseif (c <= 8e+208)
		tmp = Float64(Float64(a * i) * b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = (c * j) * t;
	tmp = 0.0;
	if (c <= -6.8e+169)
		tmp = t_1;
	elseif (c <= -3.1e-242)
		tmp = (z * x) * y;
	elseif (c <= 8e+208)
		tmp = (a * i) * b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(c * j), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[c, -6.8e+169], t$95$1, If[LessEqual[c, -3.1e-242], N[(N[(z * x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[c, 8e+208], N[(N[(a * i), $MachinePrecision] * b), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if c < -6.80000000000000056e169 or 7.9999999999999999e208 < c

   1. Initial program 45.4%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 56.6

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

   5. Applied rewrites 56.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(c \cdot j\right) \cdot t \]
   7. Step-by-step derivation

      1. lower-*.f64 48.2

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

   8. Applied rewrites 48.2%

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

   if -6.80000000000000056e169 < c < -3.10000000000000015e-242

   1. Initial program 77.1%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 52.3

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

   5. Applied rewrites 52.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \left(x \cdot z\right) \cdot y \]
   7. Step-by-step derivation

      1. *-commutative N/A

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

      2. lift-*.f64 29.6

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

   8. Applied rewrites 29.6%

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

   if -3.10000000000000015e-242 < c < 7.9999999999999999e208

   1. Initial program 73.6%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 47.9

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in z around 0

      \[\leadsto \left(a \cdot i\right) \cdot b \]
   7. Step-by-step derivation

      1. lower-*.f64 32.3

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

   8. Applied rewrites 32.3%

      \[\leadsto \left(a \cdot i\right) \cdot b \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 17: 29.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+100} \lor \neg \left(t \leq 4 \cdot 10^{+32}\right):\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot z\right) \cdot c\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= t -2.1e+100) (not (<= t 4e+32)))
   (* (- a) (* t x))
   (* (* (- b) z) c)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((t <= -2.1e+100) || !(t <= 4e+32)) {
		tmp = -a * (t * x);
	} else {
		tmp = (-b * z) * c;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b, c, i, j)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: i
    real(8), intent (in) :: j
    real(8) :: tmp
    if ((t <= (-2.1d+100)) .or. (.not. (t <= 4d+32))) then
        tmp = -a * (t * x)
    else
        tmp = (-b * z) * c
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((t <= -2.1e+100) || !(t <= 4e+32)) {
		tmp = -a * (t * x);
	} else {
		tmp = (-b * z) * c;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (t <= -2.1e+100) or not (t <= 4e+32):
		tmp = -a * (t * x)
	else:
		tmp = (-b * z) * c
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((t <= -2.1e+100) || !(t <= 4e+32))
		tmp = Float64(Float64(-a) * Float64(t * x));
	else
		tmp = Float64(Float64(Float64(-b) * z) * c);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((t <= -2.1e+100) || ~((t <= 4e+32)))
		tmp = -a * (t * x);
	else
		tmp = (-b * z) * c;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[t, -2.1e+100], N[Not[LessEqual[t, 4e+32]], $MachinePrecision]], N[((-a) * N[(t * x), $MachinePrecision]), $MachinePrecision], N[(N[((-b) * z), $MachinePrecision] * c), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.0999999999999999e100 or 4.00000000000000021e32 < t

   1. Initial program 59.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 47.2

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

   8. Applied rewrites 47.2%

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

   if -2.0999999999999999e100 < t < 4.00000000000000021e32

   1. Initial program 76.2%

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

   3. Taylor expanded in y around -inf

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

      1. associate-*r* N/A

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

      2. lower-*.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 73.1

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

   5. Applied rewrites 73.1%

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

   6. Taylor expanded in c around inf

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

      1. *-commutative N/A

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

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

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

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

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. lift-fma.f64 N/A

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

      10. lift-neg.f64 N/A

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

      11. lift-*.f64 41.5

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

   8. Applied rewrites 41.5%

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

   9. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot \left(b \cdot z\right)\right) \cdot c \]
   10. Step-by-step derivation

       1. mul-1-neg N/A

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

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

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

       3. lower-*.f64 N/A

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

       4. lift-neg.f64 35.8

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

   11. Applied rewrites 35.8%

       \[\leadsto \left(\left(-b\right) \cdot z\right) \cdot c \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.1 \cdot 10^{+100} \lor \neg \left(t \leq 4 \cdot 10^{+32}\right):\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-b\right) \cdot z\right) \cdot c\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 29.5% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+100} \lor \neg \left(t \leq 4 \cdot 10^{+32}\right):\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= t -1.8e+100) (not (<= t 4e+32)))
   (* (- a) (* t x))
   (* (* (- c) z) b)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((t <= -1.8e+100) || !(t <= 4e+32)) {
		tmp = -a * (t * x);
	} 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 ((t <= (-1.8d+100)) .or. (.not. (t <= 4d+32))) then
        tmp = -a * (t * x)
    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 ((t <= -1.8e+100) || !(t <= 4e+32)) {
		tmp = -a * (t * x);
	} else {
		tmp = (-c * z) * b;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (t <= -1.8e+100) or not (t <= 4e+32):
		tmp = -a * (t * x)
	else:
		tmp = (-c * z) * b
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((t <= -1.8e+100) || !(t <= 4e+32))
		tmp = Float64(Float64(-a) * Float64(t * x));
	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 ((t <= -1.8e+100) || ~((t <= 4e+32)))
		tmp = -a * (t * x);
	else
		tmp = (-c * z) * b;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[t, -1.8e+100], N[Not[LessEqual[t, 4e+32]], $MachinePrecision]], N[((-a) * N[(t * x), $MachinePrecision]), $MachinePrecision], N[(N[((-c) * z), $MachinePrecision] * b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.8e100 or 4.00000000000000021e32 < t

   1. Initial program 59.6%

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

   3. Taylor expanded in t around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 61.8

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

   5. Applied rewrites 61.8%

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

   6. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 47.2

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

   8. Applied rewrites 47.2%

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

   if -1.8e100 < t < 4.00000000000000021e32

   1. Initial program 76.2%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 52.7

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

   5. Applied rewrites 52.7%

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

   6. Taylor expanded in z around inf

      \[\leadsto \left(-1 \cdot \left(c \cdot z\right)\right) \cdot b \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lift-neg.f64 35.3

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

   8. Applied rewrites 35.3%

      \[\leadsto \left(\left(-c\right) \cdot z\right) \cdot b \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.8 \cdot 10^{+100} \lor \neg \left(t \leq 4 \cdot 10^{+32}\right):\\ \;\;\;\;\left(-a\right) \cdot \left(t \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-c\right) \cdot z\right) \cdot b\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 41.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;j \leq 2.7 \cdot 10^{+137}:\\ \;\;\;\;\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b\\ \mathbf{else}:\\ \;\;\;\;\left(\left(-i\right) \cdot j\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (<= j 2.7e+137) (* (fma i a (* (- c) z)) b) (* (* (- i) j) y)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if (j <= 2.7e+137) {
		tmp = fma(i, a, (-c * z)) * b;
	} else {
		tmp = (-i * j) * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if (j <= 2.7e+137)
		tmp = Float64(fma(i, a, Float64(Float64(-c) * z)) * b);
	else
		tmp = Float64(Float64(Float64(-i) * j) * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[LessEqual[j, 2.7e+137], N[(N[(i * a + N[((-c) * z), $MachinePrecision]), $MachinePrecision] * b), $MachinePrecision], N[(N[((-i) * j), $MachinePrecision] * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if j < 2.70000000000000017e137

   1. Initial program 69.4%

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

   3. Taylor expanded in b around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

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

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-*.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 49.7

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

   5. Applied rewrites 49.7%

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

   if 2.70000000000000017e137 < j

   1. Initial program 71.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 62.6

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

   5. Applied rewrites 62.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto \left(-1 \cdot \left(i \cdot j\right)\right) \cdot y \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lift-neg.f64 55.8

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

   8. Applied rewrites 55.8%

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

Alternative 20: 29.9% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+20} \lor \neg \left(y \leq 1.05 \cdot 10^{+74}\right):\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= y -5e+20) (not (<= y 1.05e+74))) (* (* z y) x) (* (* b a) i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((y <= -5e+20) || !(y <= 1.05e+74)) {
		tmp = (z * y) * x;
	} else {
		tmp = (b * a) * i;
	}
	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 ((y <= (-5d+20)) .or. (.not. (y <= 1.05d+74))) then
        tmp = (z * y) * x
    else
        tmp = (b * a) * i
    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 ((y <= -5e+20) || !(y <= 1.05e+74)) {
		tmp = (z * y) * x;
	} else {
		tmp = (b * a) * i;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (y <= -5e+20) or not (y <= 1.05e+74):
		tmp = (z * y) * x
	else:
		tmp = (b * a) * i
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((y <= -5e+20) || !(y <= 1.05e+74))
		tmp = Float64(Float64(z * y) * x);
	else
		tmp = Float64(Float64(b * a) * i);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((y <= -5e+20) || ~((y <= 1.05e+74)))
		tmp = (z * y) * x;
	else
		tmp = (b * a) * i;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[y, -5e+20], N[Not[LessEqual[y, 1.05e+74]], $MachinePrecision]], N[(N[(z * y), $MachinePrecision] * x), $MachinePrecision], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5e20 or 1.0499999999999999e74 < y

   1. Initial program 61.2%

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

   3. Taylor expanded in c around 0

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

      1. associate--l+ N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

   5. Applied rewrites 64.7%

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

   6. Taylor expanded in i around 0

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

      1. lower-fma.f64 N/A

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

      2. mul-1-neg N/A

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

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

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

      4. lower-fma.f64 N/A

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

      5. lift-neg.f64 N/A

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

      6. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(-b, z, j \cdot t\right), x \cdot \left(-1 \cdot \left(a \cdot t\right) + y \cdot z\right)\right) \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(-b, z, j \cdot t\right), x \cdot \left(-1 \cdot \left(a \cdot t\right) + y \cdot z\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(-b, z, j \cdot t\right), x \cdot \left(\left(\mathsf{neg}\left(a \cdot t\right)\right) + y \cdot z\right)\right) \]

      9. distribute-lft-neg-out N/A

         \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(-b, z, j \cdot t\right), x \cdot \left(\left(\mathsf{neg}\left(a\right)\right) \cdot t + y \cdot z\right)\right) \]

      10. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(-b, z, j \cdot t\right), x \cdot \mathsf{fma}\left(\mathsf{neg}\left(a\right), t, y \cdot z\right)\right) \]

      11. lift-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(-b, z, j \cdot t\right), x \cdot \mathsf{fma}\left(-a, t, y \cdot z\right)\right) \]

      12. lower-*.f64 51.8

          \[\leadsto \mathsf{fma}\left(c, \mathsf{fma}\left(-b, z, j \cdot t\right), x \cdot \mathsf{fma}\left(-a, t, y \cdot z\right)\right) \]

   8. Applied rewrites 51.8%

      \[\leadsto \mathsf{fma}\left(c, \color{blue}{\mathsf{fma}\left(-b, z, j \cdot t\right)}, x \cdot \mathsf{fma}\left(-a, t, y \cdot z\right)\right) \]

   9. Taylor expanded in y around inf

      \[\leadsto x \cdot \left(y \cdot \color{blue}{z}\right) \]
   10. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \left(y \cdot z\right) \cdot x \]

       2. lower-*.f64 N/A

          \[\leadsto \left(y \cdot z\right) \cdot x \]

       3. *-commutative N/A

          \[\leadsto \left(z \cdot y\right) \cdot x \]

       4. lower-*.f64 41.2

          \[\leadsto \left(z \cdot y\right) \cdot x \]

   11. Applied rewrites 41.2%

       \[\leadsto \left(z \cdot y\right) \cdot x \]

   if -5e20 < y < 1.0499999999999999e74

   1. Initial program 74.7%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot i - c \cdot z\right) \cdot \color{blue}{b} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot i - c \cdot z\right) \cdot \color{blue}{b} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(a \cdot i + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      4. *-commutative N/A

         \[\leadsto \left(i \cdot a + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, a, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, a, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, a, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, a, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      9. lower-neg.f64 53.7

         \[\leadsto \mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b \]

   5. Applied rewrites 53.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b} \]

   6. Taylor expanded in z around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto a \cdot \left(b \cdot \color{blue}{i}\right) \]

      2. lower-*.f64 25.3

         \[\leadsto a \cdot \left(b \cdot i\right) \]

   8. Applied rewrites 25.3%

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto a \cdot \left(b \cdot \color{blue}{i}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto a \cdot \left(b \cdot i\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(a \cdot b\right) \cdot i \]

      4. lower-*.f64 N/A

         \[\leadsto \left(a \cdot b\right) \cdot i \]

      5. *-commutative N/A

         \[\leadsto \left(b \cdot a\right) \cdot i \]

      6. lower-*.f64 26.9

         \[\leadsto \left(b \cdot a\right) \cdot i \]

   10. Applied rewrites 26.9%

       \[\leadsto \left(b \cdot a\right) \cdot i \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+20} \lor \neg \left(y \leq 1.05 \cdot 10^{+74}\right):\\ \;\;\;\;\left(z \cdot y\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \end{array} \]
5. Add Preprocessing

Alternative 21: 29.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;i \leq -6 \cdot 10^{-23} \lor \neg \left(i \leq 1.32 \cdot 10^{+115}\right):\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(j \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (if (or (<= i -6e-23) (not (<= i 1.32e+115))) (* (* b a) i) (* c (* j t))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double tmp;
	if ((i <= -6e-23) || !(i <= 1.32e+115)) {
		tmp = (b * a) * i;
	} else {
		tmp = c * (j * t);
	}
	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 ((i <= (-6d-23)) .or. (.not. (i <= 1.32d+115))) then
        tmp = (b * a) * i
    else
        tmp = c * (j * t)
    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 ((i <= -6e-23) || !(i <= 1.32e+115)) {
		tmp = (b * a) * i;
	} else {
		tmp = c * (j * t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	tmp = 0
	if (i <= -6e-23) or not (i <= 1.32e+115):
		tmp = (b * a) * i
	else:
		tmp = c * (j * t)
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0
	if ((i <= -6e-23) || !(i <= 1.32e+115))
		tmp = Float64(Float64(b * a) * i);
	else
		tmp = Float64(c * Float64(j * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	tmp = 0.0;
	if ((i <= -6e-23) || ~((i <= 1.32e+115)))
		tmp = (b * a) * i;
	else
		tmp = c * (j * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := If[Or[LessEqual[i, -6e-23], N[Not[LessEqual[i, 1.32e+115]], $MachinePrecision]], N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision], N[(c * N[(j * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if i < -6.00000000000000006e-23 or 1.32e115 < i

   1. Initial program 56.4%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf

      \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(a \cdot i - c \cdot z\right) \cdot \color{blue}{b} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(a \cdot i - c \cdot z\right) \cdot \color{blue}{b} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \left(a \cdot i + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      4. *-commutative N/A

         \[\leadsto \left(i \cdot a + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, a, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      6. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, a, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

      7. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(i, a, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(i, a, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

      9. lower-neg.f64 54.2

         \[\leadsto \mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b \]

   5. Applied rewrites 54.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b} \]

   6. Taylor expanded in z around 0

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto a \cdot \left(b \cdot \color{blue}{i}\right) \]

      2. lower-*.f64 41.7

         \[\leadsto a \cdot \left(b \cdot i\right) \]

   8. Applied rewrites 41.7%

      \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto a \cdot \left(b \cdot \color{blue}{i}\right) \]

      2. lift-*.f64 N/A

         \[\leadsto a \cdot \left(b \cdot i\right) \]

      3. associate-*r* N/A

         \[\leadsto \left(a \cdot b\right) \cdot i \]

      4. lower-*.f64 N/A

         \[\leadsto \left(a \cdot b\right) \cdot i \]

      5. *-commutative N/A

         \[\leadsto \left(b \cdot a\right) \cdot i \]

      6. lower-*.f64 43.4

         \[\leadsto \left(b \cdot a\right) \cdot i \]

   10. Applied rewrites 43.4%

       \[\leadsto \left(b \cdot a\right) \cdot i \]

   if -6.00000000000000006e-23 < i < 1.32e115

   1. Initial program 77.0%

      \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right)} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot \color{blue}{t} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(-1 \cdot \left(a \cdot x\right) + c \cdot j\right) \cdot \color{blue}{t} \]

      3. associate-*r* N/A

         \[\leadsto \left(\left(-1 \cdot a\right) \cdot x + c \cdot j\right) \cdot t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(a\right)\right) \cdot x + c \cdot j\right) \cdot t \]

      5. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(a\right), x, c \cdot j\right) \cdot t \]

      6. lower-neg.f64 N/A

         \[\leadsto \mathsf{fma}\left(-a, x, c \cdot j\right) \cdot t \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t \]

      8. lower-*.f64 44.5

         \[\leadsto \mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t \]

   5. Applied rewrites 44.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-a, x, j \cdot c\right) \cdot t} \]

   6. Taylor expanded in x around 0

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto c \cdot \left(j \cdot \color{blue}{t}\right) \]

      2. lower-*.f64 22.8

         \[\leadsto c \cdot \left(j \cdot t\right) \]

   8. Applied rewrites 22.8%

      \[\leadsto c \cdot \color{blue}{\left(j \cdot t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 30.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \leq -6 \cdot 10^{-23} \lor \neg \left(i \leq 1.32 \cdot 10^{+115}\right):\\ \;\;\;\;\left(b \cdot a\right) \cdot i\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(j \cdot t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 22: 22.4% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \left(b \cdot a\right) \cdot i \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* (* b a) i))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return (b * a) * 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 = (b * a) * 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 (b * a) * i;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return (b * a) * i

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(Float64(b * a) * i)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = (b * a) * i;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(N[(b * a), $MachinePrecision] * i), $MachinePrecision]

Derivation

1. Initial program 69.7%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(a \cdot i - c \cdot z\right) \cdot \color{blue}{b} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(a \cdot i - c \cdot z\right) \cdot \color{blue}{b} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \left(a \cdot i + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

   4. *-commutative N/A

      \[\leadsto \left(i \cdot a + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(i, a, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(i, a, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(i, a, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(i, a, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

   9. lower-neg.f64 44.5

      \[\leadsto \mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b \]

5. Applied rewrites 44.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b} \]

6. Taylor expanded in z around 0

   \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto a \cdot \left(b \cdot \color{blue}{i}\right) \]

   2. lower-*.f64 21.2

      \[\leadsto a \cdot \left(b \cdot i\right) \]

8. Applied rewrites 21.2%

   \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto a \cdot \left(b \cdot \color{blue}{i}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto a \cdot \left(b \cdot i\right) \]

   3. associate-*r* N/A

      \[\leadsto \left(a \cdot b\right) \cdot i \]

   4. lower-*.f64 N/A

      \[\leadsto \left(a \cdot b\right) \cdot i \]

   5. *-commutative N/A

      \[\leadsto \left(b \cdot a\right) \cdot i \]

   6. lower-*.f64 21.9

      \[\leadsto \left(b \cdot a\right) \cdot i \]

10. Applied rewrites 21.9%

    \[\leadsto \left(b \cdot a\right) \cdot i \]
11. Add Preprocessing

Alternative 23: 22.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ a \cdot \left(b \cdot i\right) \end{array} \]

FPCore

(FPCore (x y z t a b c i j) :precision binary64 (* a (* b i)))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	return a * (b * 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 = a * (b * 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 a * (b * i);
}

Python

def code(x, y, z, t, a, b, c, i, j):
	return a * (b * i)

Julia

function code(x, y, z, t, a, b, c, i, j)
	return Float64(a * Float64(b * i))
end

MATLAB

function tmp = code(x, y, z, t, a, b, c, i, j)
	tmp = a * (b * i);
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := N[(a * N[(b * i), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 69.7%

   \[\left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + j \cdot \left(c \cdot t - i \cdot y\right) \]
2. Add Preprocessing

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{b \cdot \left(a \cdot i - c \cdot z\right)} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(a \cdot i - c \cdot z\right) \cdot \color{blue}{b} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(a \cdot i - c \cdot z\right) \cdot \color{blue}{b} \]

   3. fp-cancel-sub-sign-inv N/A

      \[\leadsto \left(a \cdot i + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

   4. *-commutative N/A

      \[\leadsto \left(i \cdot a + \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

   5. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(i, a, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

   6. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(i, a, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(i, a, \left(-1 \cdot c\right) \cdot z\right) \cdot b \]

   8. mul-1-neg N/A

      \[\leadsto \mathsf{fma}\left(i, a, \left(\mathsf{neg}\left(c\right)\right) \cdot z\right) \cdot b \]

   9. lower-neg.f64 44.5

      \[\leadsto \mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b \]

5. Applied rewrites 44.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(i, a, \left(-c\right) \cdot z\right) \cdot b} \]

6. Taylor expanded in z around 0

   \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
7. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto a \cdot \left(b \cdot \color{blue}{i}\right) \]

   2. lower-*.f64 21.2

      \[\leadsto a \cdot \left(b \cdot i\right) \]

8. Applied rewrites 21.2%

   \[\leadsto a \cdot \color{blue}{\left(b \cdot i\right)} \]
9. Add Preprocessing

Developer Target 1: 68.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot \left(y \cdot z - t \cdot a\right) - b \cdot \left(c \cdot z - i \cdot a\right)\right) + \frac{j \cdot \left({\left(c \cdot t\right)}^{2} - {\left(i \cdot y\right)}^{2}\right)}{c \cdot t + i \cdot y}\\ t_2 := x \cdot \left(z \cdot y - a \cdot t\right) - \left(b \cdot \left(z \cdot c - a \cdot i\right) - \left(c \cdot t - y \cdot i\right) \cdot j\right)\\ \mathbf{if}\;t < -8.120978919195912 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < -4.712553818218485 \cdot 10^{-169}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < -7.633533346031584 \cdot 10^{-308}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t < 1.0535888557455487 \cdot 10^{-139}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c i j)
 :precision binary64
 (let* ((t_1
         (+
          (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a))))
          (/
           (* j (- (pow (* c t) 2.0) (pow (* i y) 2.0)))
           (+ (* c t) (* i y)))))
        (t_2
         (-
          (* x (- (* z y) (* a t)))
          (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j)))))
   (if (< t -8.120978919195912e-33)
     t_2
     (if (< t -4.712553818218485e-169)
       t_1
       (if (< t -7.633533346031584e-308)
         t_2
         (if (< t 1.0535888557455487e-139) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a, double b, double c, double i, double j) {
	double t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (pow((c * t), 2.0) - pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ** 2.0d0) - ((i * y) ** 2.0d0))) / ((c * t) + (i * y)))
    t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
    if (t < (-8.120978919195912d-33)) then
        tmp = t_2
    else if (t < (-4.712553818218485d-169)) then
        tmp = t_1
    else if (t < (-7.633533346031584d-308)) then
        tmp = t_2
    else if (t < 1.0535888557455487d-139) then
        tmp = t_1
    else
        tmp = t_2
    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) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (Math.pow((c * t), 2.0) - Math.pow((i * y), 2.0))) / ((c * t) + (i * y)));
	double t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	double tmp;
	if (t < -8.120978919195912e-33) {
		tmp = t_2;
	} else if (t < -4.712553818218485e-169) {
		tmp = t_1;
	} else if (t < -7.633533346031584e-308) {
		tmp = t_2;
	} else if (t < 1.0535888557455487e-139) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b, c, i, j):
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (math.pow((c * t), 2.0) - math.pow((i * y), 2.0))) / ((c * t) + (i * y)))
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j))
	tmp = 0
	if t < -8.120978919195912e-33:
		tmp = t_2
	elif t < -4.712553818218485e-169:
		tmp = t_1
	elif t < -7.633533346031584e-308:
		tmp = t_2
	elif t < 1.0535888557455487e-139:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a, b, c, i, j)
	t_1 = Float64(Float64(Float64(x * Float64(Float64(y * z) - Float64(t * a))) - Float64(b * Float64(Float64(c * z) - Float64(i * a)))) + Float64(Float64(j * Float64((Float64(c * t) ^ 2.0) - (Float64(i * y) ^ 2.0))) / Float64(Float64(c * t) + Float64(i * y))))
	t_2 = Float64(Float64(x * Float64(Float64(z * y) - Float64(a * t))) - Float64(Float64(b * Float64(Float64(z * c) - Float64(a * i))) - Float64(Float64(Float64(c * t) - Float64(y * i)) * j)))
	tmp = 0.0
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b, c, i, j)
	t_1 = ((x * ((y * z) - (t * a))) - (b * ((c * z) - (i * a)))) + ((j * (((c * t) ^ 2.0) - ((i * y) ^ 2.0))) / ((c * t) + (i * y)));
	t_2 = (x * ((z * y) - (a * t))) - ((b * ((z * c) - (a * i))) - (((c * t) - (y * i)) * j));
	tmp = 0.0;
	if (t < -8.120978919195912e-33)
		tmp = t_2;
	elseif (t < -4.712553818218485e-169)
		tmp = t_1;
	elseif (t < -7.633533346031584e-308)
		tmp = t_2;
	elseif (t < 1.0535888557455487e-139)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_, i_, j_] := Block[{t$95$1 = N[(N[(N[(x * N[(N[(y * z), $MachinePrecision] - N[(t * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(b * N[(N[(c * z), $MachinePrecision] - N[(i * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(j * N[(N[Power[N[(c * t), $MachinePrecision], 2.0], $MachinePrecision] - N[Power[N[(i * y), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(c * t), $MachinePrecision] + N[(i * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * N[(N[(z * y), $MachinePrecision] - N[(a * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(b * N[(N[(z * c), $MachinePrecision] - N[(a * i), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * t), $MachinePrecision] - N[(y * i), $MachinePrecision]), $MachinePrecision] * j), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -8.120978919195912e-33], t$95$2, If[Less[t, -4.712553818218485e-169], t$95$1, If[Less[t, -7.633533346031584e-308], t$95$2, If[Less[t, 1.0535888557455487e-139], t$95$1, t$95$2]]]]]]

Reproduce

herbie shell --seed 2025051 
(FPCore (x y z t a b c i j)
  :name "Linear.Matrix:det33 from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -1015122364899489/125000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t -942510763643697/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (if (< t -238547917063487/3125000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))) (if (< t 10535888557455487/100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (/ (* j (- (pow (* c t) 2) (pow (* i y) 2))) (+ (* c t) (* i y)))) (- (* x (- (* z y) (* a t))) (- (* b (- (* z c) (* a i))) (* (- (* c t) (* y i)) j))))))))

  (+ (- (* x (- (* y z) (* t a))) (* b (- (* c z) (* i a)))) (* j (- (* c t) (* i y)))))