Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.7% → 97.6%

Time: 4.7s

Alternatives: 8

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(y + z\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (+ y z)) z))

C

double code(double x, double y, double z) {
	return (x * (y + z)) / z;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (y + z)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) / z;
}

Python

def code(x, y, z):
	return (x * (y + z)) / z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y + z)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(y + z\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (+ y z)) z))

C

double code(double x, double y, double z) {
	return (x * (y + z)) / z;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x * (y + z)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) / z;
}

Python

def code(x, y, z):
	return (x * (y + z)) / z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y + z)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{z}, y \cdot x\_m, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x\_m, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1e-69) (fma (/ 1.0 z) (* y x_m) x_m) (fma (/ y z) x_m x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1e-69) {
		tmp = fma((1.0 / z), (y * x_m), x_m);
	} else {
		tmp = fma((y / z), x_m, x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 1e-69)
		tmp = fma(Float64(1.0 / z), Float64(y * x_m), x_m);
	else
		tmp = fma(Float64(y / z), x_m, x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1e-69], N[(N[(1.0 / z), $MachinePrecision] * N[(y * x$95$m), $MachinePrecision] + x$95$m), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.9999999999999996e-70

   1. Initial program 91.3%

      \[\frac{x \cdot \left(y + z\right)}{z} \]

   3. Applied rewrites 95.0%

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

   if 9.9999999999999996e-70 < x

   1. Initial program 80.6%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-lft-out N/A

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

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + x \cdot z}{z}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot z + x \cdot y}}{z} \]

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

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

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

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

      10. div-sub N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

          \[\leadsto x \cdot \color{blue}{1} - \frac{\mathsf{neg}\left(x \cdot y\right)}{z} \]

      13. distribute-neg-frac N/A

          \[\leadsto x \cdot 1 - \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y}{z}\right)\right)} \]

      14. mul-1-neg N/A

          \[\leadsto x \cdot 1 - \color{blue}{-1 \cdot \frac{x \cdot y}{z}} \]

      15. *-rgt-identity N/A

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

      16. distribute-rgt-out-- N/A

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

      17. mul-1-neg N/A

          \[\leadsto 1 \cdot \left(x - \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y}{z}\right)\right)}\right) \]

      18. associate-/l* N/A

          \[\leadsto 1 \cdot \left(x - \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{y}{z}}\right)\right)\right) \]

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

          \[\leadsto 1 \cdot \left(x - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{z}}\right) \]

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

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

      21. associate-/l* N/A

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

      22. *-lft-identity N/A

          \[\leadsto \color{blue}{x + \frac{x \cdot y}{z}} \]

      23. +-commutative N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{z} + x} \]

   3. Applied rewrites 99.3%

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

Alternative 2: 97.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y}{z}, x\_m, x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.5 \cdot 10^{-251}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x\_m \leq 2.95 \cdot 10^{-126}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{z}, y, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (fma (/ y z) x_m x_m)))
   (*
    x_s
    (if (<= x_m 3.5e-251)
      t_0
      (if (<= x_m 2.95e-126) (fma (/ x_m z) y x_m) t_0)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = fma((y / z), x_m, x_m);
	double tmp;
	if (x_m <= 3.5e-251) {
		tmp = t_0;
	} else if (x_m <= 2.95e-126) {
		tmp = fma((x_m / z), y, x_m);
	} else {
		tmp = t_0;
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = fma(Float64(y / z), x_m, x_m)
	tmp = 0.0
	if (x_m <= 3.5e-251)
		tmp = t_0;
	elseif (x_m <= 2.95e-126)
		tmp = fma(Float64(x_m / z), y, x_m);
	else
		tmp = t_0;
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(y / z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 3.5e-251], t$95$0, If[LessEqual[x$95$m, 2.95e-126], N[(N[(x$95$m / z), $MachinePrecision] * y + x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.50000000000000034e-251 or 2.94999999999999986e-126 < x

   1. Initial program 82.9%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-lft-out N/A

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

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + x \cdot z}{z}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot z + x \cdot y}}{z} \]

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

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

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

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

      10. div-sub N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

          \[\leadsto x \cdot \color{blue}{1} - \frac{\mathsf{neg}\left(x \cdot y\right)}{z} \]

      13. distribute-neg-frac N/A

          \[\leadsto x \cdot 1 - \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y}{z}\right)\right)} \]

      14. mul-1-neg N/A

          \[\leadsto x \cdot 1 - \color{blue}{-1 \cdot \frac{x \cdot y}{z}} \]

      15. *-rgt-identity N/A

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

      16. distribute-rgt-out-- N/A

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

      17. mul-1-neg N/A

          \[\leadsto 1 \cdot \left(x - \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y}{z}\right)\right)}\right) \]

      18. associate-/l* N/A

          \[\leadsto 1 \cdot \left(x - \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{y}{z}}\right)\right)\right) \]

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

          \[\leadsto 1 \cdot \left(x - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{z}}\right) \]

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

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

      21. associate-/l* N/A

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

      22. *-lft-identity N/A

          \[\leadsto \color{blue}{x + \frac{x \cdot y}{z}} \]

      23. +-commutative N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{z} + x} \]

   3. Applied rewrites 97.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z}, x, x\right)} \]

   if 3.50000000000000034e-251 < x < 2.94999999999999986e-126

   1. Initial program 91.7%

      \[\frac{x \cdot \left(y + z\right)}{z} \]

   3. Applied rewrites 95.2%

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

Alternative 3: 96.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 9 \cdot 10^{-70}:\\ \;\;\;\;\frac{x\_m \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x\_m, x\_m\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= x_m 9e-70) (/ (* x_m (+ y z)) z) (fma (/ y z) x_m x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 9e-70) {
		tmp = (x_m * (y + z)) / z;
	} else {
		tmp = fma((y / z), x_m, x_m);
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 9e-70)
		tmp = Float64(Float64(x_m * Float64(y + z)) / z);
	else
		tmp = fma(Float64(y / z), x_m, x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 9e-70], N[(N[(x$95$m * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 9.00000000000000044e-70

   1. Initial program 91.3%

      \[\frac{x \cdot \left(y + z\right)}{z} \]

   if 9.00000000000000044e-70 < x

   1. Initial program 80.6%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. distribute-lft-out N/A

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

      6. frac-2neg N/A

         \[\leadsto \color{blue}{\frac{x \cdot y + x \cdot z}{z}} \]

      7. +-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot z + x \cdot y}}{z} \]

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

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

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

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

      10. div-sub N/A

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

      11. associate-/l* N/A

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

      12. *-inverses N/A

          \[\leadsto x \cdot \color{blue}{1} - \frac{\mathsf{neg}\left(x \cdot y\right)}{z} \]

      13. distribute-neg-frac N/A

          \[\leadsto x \cdot 1 - \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y}{z}\right)\right)} \]

      14. mul-1-neg N/A

          \[\leadsto x \cdot 1 - \color{blue}{-1 \cdot \frac{x \cdot y}{z}} \]

      15. *-rgt-identity N/A

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

      16. distribute-rgt-out-- N/A

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

      17. mul-1-neg N/A

          \[\leadsto 1 \cdot \left(x - \color{blue}{\left(\mathsf{neg}\left(\frac{x \cdot y}{z}\right)\right)}\right) \]

      18. associate-/l* N/A

          \[\leadsto 1 \cdot \left(x - \left(\mathsf{neg}\left(\color{blue}{x \cdot \frac{y}{z}}\right)\right)\right) \]

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

          \[\leadsto 1 \cdot \left(x - \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y}{z}}\right) \]

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

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

      21. associate-/l* N/A

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

      22. *-lft-identity N/A

          \[\leadsto \color{blue}{x + \frac{x \cdot y}{z}} \]

      23. +-commutative N/A

          \[\leadsto \color{blue}{\frac{x \cdot y}{z} + x} \]

   3. Applied rewrites 99.3%

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

Alternative 4: 94.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \mathsf{fma}\left(\frac{x\_m}{z}, y, x\_m\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (fma (/ x_m z) y x_m)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * fma((x_m / z), y, x_m);
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * fma(Float64(x_m / z), y, x_m))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * N[(N[(x$95$m / z), $MachinePrecision] * y + x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.7%

   \[\frac{x \cdot \left(y + z\right)}{z} \]

3. Applied rewrites 94.1%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{z}, y, x\right)} \]
4. Add Preprocessing

Alternative 5: 72.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+22}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+48}:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= z -1e+22) x_m (if (<= z 2.4e+48) (/ (* x_m y) z) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -1e+22) {
		tmp = x_m;
	} else if (z <= 2.4e+48) {
		tmp = (x_m * y) / z;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1d+22)) then
        tmp = x_m
    else if (z <= 2.4d+48) then
        tmp = (x_m * y) / z
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -1e+22) {
		tmp = x_m;
	} else if (z <= 2.4e+48) {
		tmp = (x_m * y) / z;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= -1e+22:
		tmp = x_m
	elif z <= 2.4e+48:
		tmp = (x_m * y) / z
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= -1e+22)
		tmp = x_m;
	elseif (z <= 2.4e+48)
		tmp = Float64(Float64(x_m * y) / z);
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= -1e+22)
		tmp = x_m;
	elseif (z <= 2.4e+48)
		tmp = (x_m * y) / z;
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -1e+22], x$95$m, If[LessEqual[z, 2.4e+48], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1e22 or 2.4000000000000001e48 < z

   1. Initial program 73.9%

      \[\frac{x \cdot \left(y + z\right)}{z} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 76.9%

      \[\leadsto \color{blue}{x} \]

   if -1e22 < z < 2.4000000000000001e48

   1. Initial program 93.5%

      \[\frac{x \cdot \left(y + z\right)}{z} \]

   2. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot \color{blue}{y}}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 69.2%

      \[\leadsto \frac{x \cdot \color{blue}{y}}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 72.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+22}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+48}:\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= z -1.25e+22) x_m (if (<= z 1.4e+48) (* (/ x_m z) y) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -1.25e+22) {
		tmp = x_m;
	} else if (z <= 1.4e+48) {
		tmp = (x_m / z) * y;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-1.25d+22)) then
        tmp = x_m
    else if (z <= 1.4d+48) then
        tmp = (x_m / z) * y
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (z <= -1.25e+22) {
		tmp = x_m;
	} else if (z <= 1.4e+48) {
		tmp = (x_m / z) * y;
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if z <= -1.25e+22:
		tmp = x_m
	elif z <= 1.4e+48:
		tmp = (x_m / z) * y
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (z <= -1.25e+22)
		tmp = x_m;
	elseif (z <= 1.4e+48)
		tmp = Float64(Float64(x_m / z) * y);
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= -1.25e+22)
		tmp = x_m;
	elseif (z <= 1.4e+48)
		tmp = (x_m / z) * y;
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -1.25e+22], x$95$m, If[LessEqual[z, 1.4e+48], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.2499999999999999e22 or 1.40000000000000006e48 < z

   1. Initial program 73.8%

      \[\frac{x \cdot \left(y + z\right)}{z} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 76.9%

      \[\leadsto \color{blue}{x} \]

   if -1.2499999999999999e22 < z < 1.40000000000000006e48

   1. Initial program 93.6%

      \[\frac{x \cdot \left(y + z\right)}{z} \]

   2. Taylor expanded in y around inf

      \[\leadsto \frac{x \cdot \color{blue}{y}}{z} \]
   3. Step-by-step derivation

   4. Applied rewrites 69.2%

      \[\leadsto \frac{x \cdot \color{blue}{y}}{z} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot x}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{x}{z}} \]

      5. *-commutative N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

      6. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]

      7. lift-/.f64 68.9

         \[\leadsto \color{blue}{\frac{x}{z}} \cdot y \]

   6. Applied rewrites 68.9%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \left(y + z\right)}{z}\\ t_1 := \frac{y}{z} \cdot x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+200}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* x_m (+ y z)) z)) (t_1 (* (/ y z) x_m)))
   (* x_s (if (<= t_0 0.0) t_1 (if (<= t_0 5e+200) x_m t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (y + z)) / z;
	double t_1 = (y / z) * x_m;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+200) {
		tmp = x_m;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_m * (y + z)) / z
    t_1 = (y / z) * x_m
    if (t_0 <= 0.0d0) then
        tmp = t_1
    else if (t_0 <= 5d+200) then
        tmp = x_m
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (y + z)) / z;
	double t_1 = (y / z) * x_m;
	double tmp;
	if (t_0 <= 0.0) {
		tmp = t_1;
	} else if (t_0 <= 5e+200) {
		tmp = x_m;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (x_m * (y + z)) / z
	t_1 = (y / z) * x_m
	tmp = 0
	if t_0 <= 0.0:
		tmp = t_1
	elif t_0 <= 5e+200:
		tmp = x_m
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m * Float64(y + z)) / z)
	t_1 = Float64(Float64(y / z) * x_m)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+200)
		tmp = x_m;
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (x_m * (y + z)) / z;
	t_1 = (y / z) * x_m;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = t_1;
	elseif (t_0 <= 5e+200)
		tmp = x_m;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] * x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 0.0], t$95$1, If[LessEqual[t$95$0, 5e+200], x$95$m, t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < 0.0 or 5.00000000000000019e200 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 71.9%

      \[\frac{x \cdot \left(y + z\right)}{z} \]

   3. Applied rewrites 88.7%

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

   4. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \]
   5. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{x} \cdot y}{z} \]

      2. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{z} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{x \cdot y}{z} \]

      4. *-lft-identity N/A

         \[\leadsto \frac{x \cdot y}{z} \]

      5. *-inverses N/A

         \[\leadsto \frac{x \cdot y}{z} \]

      6. div-add N/A

         \[\leadsto \frac{\color{blue}{x} \cdot y}{z} \]

      7. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{x \cdot y}}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x} \cdot y}{z} \]

      9. *-rgt-identity N/A

         \[\leadsto \frac{\color{blue}{x} \cdot y}{z} \]

      10. *-inverses N/A

          \[\leadsto \frac{x \cdot y}{z} \]

      11. associate-/l* N/A

          \[\leadsto \frac{\color{blue}{x} \cdot y}{z} \]

      12. +-commutative N/A

          \[\leadsto \frac{x \cdot y}{z} \]

      13. *-commutative N/A

          \[\leadsto \frac{x \cdot y}{z} \]

      14. *-commutative N/A

          \[\leadsto \frac{x \cdot y}{z} \]

      15. associate-/r* N/A

          \[\leadsto \frac{\color{blue}{x \cdot y}}{z} \]

      16. associate-/l* N/A

          \[\leadsto x \cdot \color{blue}{\frac{y}{z}} \]

      17. *-commutative N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{x} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{y}{z} \cdot \color{blue}{x} \]

      19. lift-/.f64 66.0

          \[\leadsto \frac{y}{z} \cdot x \]

   6. Applied rewrites 66.0%

      \[\leadsto \color{blue}{\frac{y}{z} \cdot x} \]

   if 0.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < 5.00000000000000019e200

   1. Initial program 99.4%

      \[\frac{x \cdot \left(y + z\right)}{z} \]

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 74.8%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 50.6% accurate, 10.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot x\_m \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s x_m))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	return x_s * x_m;
}

Fortran

x\_m =     private
x\_s =     private
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_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * x_m
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	return x_s * x_m;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	return x_s * x_m

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * x_m)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z)
	tmp = x_s * x_m;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 84.7%

   \[\frac{x \cdot \left(y + z\right)}{z} \]

2. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 50.6%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025130 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64
  (/ (* x (+ y z)) z))