Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.0% → 97.7%

Time: 5.9s

Alternatives: 7

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: 84.0% 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: 97.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \left(y + z\right)}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{-47} \lor \neg \left(t\_0 \leq 2 \cdot 10^{+292}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x\_m, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x\_m, 1, \frac{y \cdot x\_m}{z}\right)\\ \end{array} \end{array} \end{array} \]

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)))
   (*
    x_s
    (if (or (<= t_0 2e-47) (not (<= t_0 2e+292)))
      (fma (/ y z) x_m x_m)
      (fma x_m 1.0 (/ (* y x_m) z))))))

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 tmp;
	if ((t_0 <= 2e-47) || !(t_0 <= 2e+292)) {
		tmp = fma((y / z), x_m, x_m);
	} else {
		tmp = fma(x_m, 1.0, ((y * x_m) / z));
	}
	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)
	tmp = 0.0
	if ((t_0 <= 2e-47) || !(t_0 <= 2e+292))
		tmp = fma(Float64(y / z), x_m, x_m);
	else
		tmp = fma(x_m, 1.0, Float64(Float64(y * x_m) / z));
	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[(x$95$m * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[Or[LessEqual[t$95$0, 2e-47], N[Not[LessEqual[t$95$0, 2e+292]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision], N[(x$95$m * 1.0 + N[(N[(y * x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < 1.9999999999999999e-47 or 2e292 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 81.1%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-lft-identity N/A

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

      17. metadata-eval N/A

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

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

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

      19. mul-1-neg N/A

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

      20. distribute-lft-out N/A

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

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

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

   5. Applied rewrites 98.5%

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

   if 1.9999999999999999e-47 < (/.f64 (*.f64 x (+.f64 y z)) z) < 2e292

   1. Initial program 99.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. +-commutative N/A

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

      5. distribute-rgt-in N/A

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

      6. div-add N/A

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

      7. *-commutative N/A

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

      8. associate-/l* N/A

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

      9. lower-fma.f64 N/A

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

      10. *-inverses N/A

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

      11. lower-/.f64 N/A

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

      12. lower-*.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 1, \frac{y \cdot x}{z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq 2 \cdot 10^{-47} \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \leq 2 \cdot 10^{+292}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \frac{y \cdot x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \left(y + z\right)}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 40:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x\_m, x\_m\right)\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+307}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{z}, y, x\_m\right)\\ \end{array} \end{array} \end{array} \]

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)))
   (*
    x_s
    (if (<= t_0 40.0)
      (fma (/ y z) x_m x_m)
      (if (<= t_0 5e+307) t_0 (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) {
	double t_0 = (x_m * (y + z)) / z;
	double tmp;
	if (t_0 <= 40.0) {
		tmp = fma((y / z), x_m, x_m);
	} else if (t_0 <= 5e+307) {
		tmp = t_0;
	} else {
		tmp = fma((x_m / z), y, 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)
	t_0 = Float64(Float64(x_m * Float64(y + z)) / z)
	tmp = 0.0
	if (t_0 <= 40.0)
		tmp = fma(Float64(y / z), x_m, x_m);
	elseif (t_0 <= 5e+307)
		tmp = t_0;
	else
		tmp = fma(Float64(x_m / z), y, 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_] := Block[{t$95$0 = N[(N[(x$95$m * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 40.0], N[(N[(y / z), $MachinePrecision] * x$95$m + x$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e+307], t$95$0, N[(N[(x$95$m / z), $MachinePrecision] * y + x$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < 40

   1. Initial program 85.7%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-lft-identity N/A

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

      17. metadata-eval N/A

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

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

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

      19. mul-1-neg N/A

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

      20. distribute-lft-out N/A

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

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

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

   5. Applied rewrites 98.2%

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

   if 40 < (/.f64 (*.f64 x (+.f64 y z)) z) < 5e307

   1. Initial program 99.7%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing

   if 5e307 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 63.5%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-lft-identity N/A

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

      17. metadata-eval N/A

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

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

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

      19. mul-1-neg N/A

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

      20. distribute-lft-out N/A

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

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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq 40:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 5 \cdot 10^{+307}:\\ \;\;\;\;\frac{x \cdot \left(y + z\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \left(y + z\right)}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0:\\ \;\;\;\;\frac{y}{z} \cdot x\_m\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+201}:\\ \;\;\;\;\frac{z \cdot x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \end{array} \end{array} \end{array} \]

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)))
   (*
    x_s
    (if (<= t_0 0.0)
      (* (/ y z) x_m)
      (if (<= t_0 5e+201) (/ (* z x_m) z) (* (/ x_m z) y))))))

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 tmp;
	if (t_0 <= 0.0) {
		tmp = (y / z) * x_m;
	} else if (t_0 <= 5e+201) {
		tmp = (z * x_m) / z;
	} else {
		tmp = (x_m / z) * y;
	}
	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) :: tmp
    t_0 = (x_m * (y + z)) / z
    if (t_0 <= 0.0d0) then
        tmp = (y / z) * x_m
    else if (t_0 <= 5d+201) then
        tmp = (z * x_m) / z
    else
        tmp = (x_m / z) * y
    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 tmp;
	if (t_0 <= 0.0) {
		tmp = (y / z) * x_m;
	} else if (t_0 <= 5e+201) {
		tmp = (z * x_m) / z;
	} else {
		tmp = (x_m / z) * y;
	}
	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
	tmp = 0
	if t_0 <= 0.0:
		tmp = (y / z) * x_m
	elif t_0 <= 5e+201:
		tmp = (z * x_m) / z
	else:
		tmp = (x_m / z) * y
	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)
	tmp = 0.0
	if (t_0 <= 0.0)
		tmp = Float64(Float64(y / z) * x_m);
	elseif (t_0 <= 5e+201)
		tmp = Float64(Float64(z * x_m) / z);
	else
		tmp = Float64(Float64(x_m / z) * y);
	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;
	tmp = 0.0;
	if (t_0 <= 0.0)
		tmp = (y / z) * x_m;
	elseif (t_0 <= 5e+201)
		tmp = (z * x_m) / z;
	else
		tmp = (x_m / z) * y;
	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]}, N[(x$95$s * If[LessEqual[t$95$0, 0.0], N[(N[(y / z), $MachinePrecision] * x$95$m), $MachinePrecision], If[LessEqual[t$95$0, 5e+201], N[(N[(z * x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 81.7%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-lft-identity N/A

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

      17. metadata-eval N/A

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

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

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

      19. mul-1-neg N/A

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

      20. distribute-lft-out N/A

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

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

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

   5. Applied rewrites 97.7%

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

   7. Applied rewrites 97.7%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 48.3%

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

   if 0.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < 4.9999999999999995e201

   1. Initial program 99.1%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 58.1

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

   5. Applied rewrites 58.1%

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

   if 4.9999999999999995e201 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 70.6%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-lft-identity N/A

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

      17. metadata-eval N/A

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

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

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

      19. mul-1-neg N/A

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

      20. distribute-lft-out N/A

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

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

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

   5. Applied rewrites 94.6%

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

   7. Applied rewrites 94.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 70.1%

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

   12. Applied rewrites 71.7%

       \[\leadsto \frac{x}{z} \cdot y \]
3. Recombined 3 regimes into one program.

4. Final simplification 55.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq 0:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;\frac{x \cdot \left(y + z\right)}{z} \leq 5 \cdot 10^{+201}:\\ \;\;\;\;\frac{z \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \left(y + z\right)}{z} \leq 2 \cdot 10^{-32}:\\ \;\;\;\;\frac{y}{z} \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \end{array} \end{array} \]

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 (+ y z)) z) 2e-32) (* (/ y z) x_m) (* (/ x_m z) y))))

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 * (y + z)) / z) <= 2e-32) {
		tmp = (y / z) * x_m;
	} else {
		tmp = (x_m / z) * y;
	}
	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 (((x_m * (y + z)) / z) <= 2d-32) then
        tmp = (y / z) * x_m
    else
        tmp = (x_m / z) * y
    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 (((x_m * (y + z)) / z) <= 2e-32) {
		tmp = (y / z) * x_m;
	} else {
		tmp = (x_m / z) * y;
	}
	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 ((x_m * (y + z)) / z) <= 2e-32:
		tmp = (y / z) * x_m
	else:
		tmp = (x_m / z) * y
	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 (Float64(Float64(x_m * Float64(y + z)) / z) <= 2e-32)
		tmp = Float64(Float64(y / z) * x_m);
	else
		tmp = Float64(Float64(x_m / z) * y);
	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 (((x_m * (y + z)) / z) <= 2e-32)
		tmp = (y / z) * x_m;
	else
		tmp = (x_m / z) * y;
	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[N[(N[(x$95$m * N[(y + z), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], 2e-32], N[(N[(y / z), $MachinePrecision] * x$95$m), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < 2.00000000000000011e-32

   1. Initial program 85.5%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-lft-identity N/A

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

      17. metadata-eval N/A

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

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

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

      19. mul-1-neg N/A

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

      20. distribute-lft-out N/A

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

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

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

   5. Applied rewrites 98.2%

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

   7. Applied rewrites 98.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 45.2%

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

   if 2.00000000000000011e-32 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 82.8%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-lft-identity N/A

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

      17. metadata-eval N/A

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

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

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

      19. mul-1-neg N/A

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

      20. distribute-lft-out N/A

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

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

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

   5. Applied rewrites 91.4%

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

   7. Applied rewrites 91.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 58.1%

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

   12. Applied rewrites 63.4%

       \[\leadsto \frac{x}{z} \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq 2 \cdot 10^{-32}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{z}, y, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x\_m, x\_m\right)\\ \end{array} \end{array} \]

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 2e-49) (fma (/ x_m z) y 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 <= 2e-49) {
		tmp = fma((x_m / z), y, 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 <= 2e-49)
		tmp = fma(Float64(x_m / z), y, 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, 2e-49], N[(N[(x$95$m / z), $MachinePrecision] * y + 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 < 1.99999999999999987e-49

   1. Initial program 86.6%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-lft-identity N/A

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

      17. metadata-eval N/A

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

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

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

      19. mul-1-neg N/A

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

      20. distribute-lft-out N/A

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

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

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

   5. Applied rewrites 94.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 93.9%

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

   if 1.99999999999999987e-49 < x

   1. Initial program 80.2%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. distribute-lft-in N/A

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

      2. +-commutative N/A

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

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

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. associate-/l* N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. associate-*l* N/A

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

      10. *-commutative N/A

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

      11. associate-*r/ N/A

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

      12. associate-*r* N/A

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

      13. associate-/l* N/A

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

      14. *-commutative N/A

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

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

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

      16. *-lft-identity N/A

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

      17. metadata-eval N/A

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

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

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

      19. mul-1-neg N/A

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

      20. distribute-lft-out N/A

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

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

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

   5. Applied rewrites 99.9%

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

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2 \cdot 10^{-49}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 94.3% 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 84.6%

   \[\frac{x \cdot \left(y + z\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

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

   2. +-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. *-inverses N/A

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

   8. metadata-eval N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. associate-*r/ N/A

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

   12. associate-*r* N/A

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

   13. associate-/l* N/A

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

   14. *-commutative N/A

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

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

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

   16. *-lft-identity N/A

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

   17. metadata-eval N/A

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

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

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

   19. mul-1-neg N/A

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

   20. distribute-lft-out N/A

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

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

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

5. Applied rewrites 95.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 94.4%

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

9. Final simplification 94.4%

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

Alternative 7: 47.3% accurate, 1.2× speedup

Math

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

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 (* (/ x_m z) y)))

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 / z) * y);
}

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 / z) * y)
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 / z) * y);
}

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 / z) * y)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(Float64(x_m / z) * y))
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 / z) * y);
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), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.6%

   \[\frac{x \cdot \left(y + z\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. distribute-lft-in N/A

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

   2. +-commutative N/A

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

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

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

   4. mul-1-neg N/A

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

   5. div-sub N/A

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

   6. associate-/l* N/A

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

   7. *-inverses N/A

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

   8. metadata-eval N/A

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

   9. associate-*l* N/A

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

   10. *-commutative N/A

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

   11. associate-*r/ N/A

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

   12. associate-*r* N/A

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

   13. associate-/l* N/A

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

   14. *-commutative N/A

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

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

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

   16. *-lft-identity N/A

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

   17. metadata-eval N/A

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

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

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

   19. mul-1-neg N/A

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

   20. distribute-lft-out N/A

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

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

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

5. Applied rewrites 95.8%

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

7. Applied rewrites 95.8%

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

8. Taylor expanded in y around inf

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

10. Applied rewrites 49.7%

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

12. Applied rewrites 50.2%

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

13. Final simplification 50.2%

    \[\leadsto \frac{x}{z} \cdot y \]
14. Add Preprocessing

Developer Target 1: 96.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{z}{y + z}} \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return x / (z / (y + 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 / (z / (y + z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025008 
(FPCore (x y z)
  :name "Numeric.SpecFunctions:choose from math-functions-0.1.5.2"
  :precision binary64

  :alt
  (! :herbie-platform default (/ x (/ z (+ y z))))

  (/ (* x (+ y z)) z))