Numeric.SpecFunctions:choose from math-functions-0.1.5.2

Percentage Accurate: 84.9% → 97.7%

Time: 5.6s

Alternatives: 5

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: 84.9% 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, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.2 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x\_m}{z}, y, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x\_m, x\_m\right)\\ \end{array} \end{array} \]

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 2.2e+50) (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 <= 2.2e+50) {
		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 <= 2.2e+50)
		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, 2.2e+50], 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 < 2.20000000000000017e50

   1. Initial program 85.3%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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. associate-/l* N/A

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

      2. +-commutative N/A

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

      3. div-add N/A

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

      4. distribute-rgt-in N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. distribute-lft-out N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. associate-/l* N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 93.9

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

   5. Applied rewrites 93.9%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 94.8

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

   8. Applied rewrites 94.8%

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

   if 2.20000000000000017e50 < x

   1. Initial program 68.0%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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. associate-/l* N/A

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

      2. +-commutative N/A

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

      3. div-add N/A

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

      4. distribute-rgt-in N/A

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

      5. *-inverses N/A

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

      6. metadata-eval N/A

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

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

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

      8. *-commutative N/A

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

      9. associate-/l* N/A

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

      10. remove-double-neg N/A

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

      11. mul-1-neg N/A

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

      12. distribute-neg-in N/A

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

      13. distribute-lft-out N/A

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

      14. mul-1-neg N/A

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

      15. remove-double-neg N/A

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

      16. +-commutative N/A

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

      17. associate-/l* N/A

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

      18. *-commutative N/A

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

      19. lower-fma.f64 N/A

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

      20. lower-/.f64 99.9

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

   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. Add Preprocessing

Alternative 2: 72.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \left(y + z\right)}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0 \lor \neg \left(t\_0 \leq 10^{+279}\right):\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot x\_m}{z}\\ \end{array} \end{array} \end{array} \]

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 0.0) (not (<= t_0 1e+279)))
      (* (/ x_m z) y)
      (/ (* z 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 <= 0.0) || !(t_0 <= 1e+279)) {
		tmp = (x_m / z) * y;
	} else {
		tmp = (z * x_m) / z;
	}
	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) .or. (.not. (t_0 <= 1d+279))) then
        tmp = (x_m / z) * y
    else
        tmp = (z * x_m) / z
    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) || !(t_0 <= 1e+279)) {
		tmp = (x_m / z) * y;
	} else {
		tmp = (z * x_m) / z;
	}
	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) or not (t_0 <= 1e+279):
		tmp = (x_m / z) * y
	else:
		tmp = (z * 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 <= 0.0) || !(t_0 <= 1e+279))
		tmp = Float64(Float64(x_m / z) * y);
	else
		tmp = Float64(Float64(z * x_m) / z);
	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) || ~((t_0 <= 1e+279)))
		tmp = (x_m / z) * y;
	else
		tmp = (z * x_m) / z;
	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[Or[LessEqual[t$95$0, 0.0], N[Not[LessEqual[t$95$0, 1e+279]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(z * x$95$m), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (+.f64 y z)) z) < -0.0 or 1.00000000000000006e279 < (/.f64 (*.f64 x (+.f64 y z)) z)

   1. Initial program 74.0%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({y}^{\color{blue}{\left(\frac{2}{2}\right)}} + z\right)}{z} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{y}^{2}}} + z\right)}{z} \]

      5. pow2 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. unpow1 N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left(\left|y\right| + {z}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}{z} \]

      9. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\left|y\right| + \color{blue}{\sqrt{{z}^{2}}}\right)}{z} \]

      10. pow2 N/A

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

      11. rem-sqrt-square-rev N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

          \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{{y}^{2}}} + \left|z\right|\right)}{z} \]

      14. sqrt-pow1 N/A

          \[\leadsto \frac{x \cdot \left(\color{blue}{{y}^{\left(\frac{2}{2}\right)}} + \left|z\right|\right)}{z} \]

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. unpow1 N/A

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

      18. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left({y}^{\color{blue}{\left(\frac{2}{2}\right)}} + \left|z\right|\right)}{z} \]

      19. sqrt-pow1 N/A

          \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{y}^{2}}} + \left|z\right|\right)}{z} \]

      20. pow2 N/A

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

      21. sqrt-prod N/A

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

      22. rem-sqrt-square-rev N/A

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

      23. pow2 N/A

          \[\leadsto \frac{x \cdot \left(\sqrt{y} \cdot \sqrt{y} + \sqrt{\color{blue}{{z}^{2}}}\right)}{z} \]

      24. sqrt-pow1 N/A

          \[\leadsto \frac{x \cdot \left(\sqrt{y} \cdot \sqrt{y} + \color{blue}{{z}^{\left(\frac{2}{2}\right)}}\right)}{z} \]

      25. metadata-eval N/A

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

      26. unpow1 N/A

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

      27. lower-fma.f64 N/A

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

      28. lower-sqrt.f64 N/A

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

      29. lower-sqrt.f64 34.0

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

   4. Applied rewrites 34.0%

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

   5. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{z}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{z}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \frac{y \cdot {\left(\sqrt{-1}\right)}^{2}}{z}}\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y \cdot {\left(\sqrt{-1}\right)}^{2}}{z}} \]

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \frac{y \cdot {\left(\sqrt{-1}\right)}^{2}}{z} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot x\right) \cdot \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot y}}{z} \]

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. associate-/l* N/A

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

      16. associate-*l/ N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 52.8

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

   7. Applied rewrites 52.8%

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

   if -0.0 < (/.f64 (*.f64 x (+.f64 y z)) z) < 1.00000000000000006e279

   1. Initial program 99.7%

      \[\frac{x \cdot \left(y + z\right)}{z} \]
   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 49.4

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

   5. Applied rewrites 49.4%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y + z\right)}{z} \leq 0 \lor \neg \left(\frac{x \cdot \left(y + z\right)}{z} \leq 10^{+279}\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 49.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \end{array} \end{array} \]

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 3.2e-18) (* (/ x_m z) y) (* x_m (/ y 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 tmp;
	if (x_m <= 3.2e-18) {
		tmp = (x_m / z) * y;
	} else {
		tmp = x_m * (y / z);
	}
	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 <= 3.2d-18) then
        tmp = (x_m / z) * y
    else
        tmp = x_m * (y / z)
    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 <= 3.2e-18) {
		tmp = (x_m / z) * y;
	} else {
		tmp = x_m * (y / z);
	}
	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 <= 3.2e-18:
		tmp = (x_m / z) * y
	else:
		tmp = x_m * (y / z)
	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 <= 3.2e-18)
		tmp = Float64(Float64(x_m / z) * y);
	else
		tmp = Float64(x_m * Float64(y / z));
	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 <= 3.2e-18)
		tmp = (x_m / z) * y;
	else
		tmp = x_m * (y / z);
	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[x$95$m, 3.2e-18], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.1999999999999999e-18

   1. Initial program 84.7%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({y}^{\color{blue}{\left(\frac{2}{2}\right)}} + z\right)}{z} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{y}^{2}}} + z\right)}{z} \]

      5. pow2 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. unpow1 N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left(\left|y\right| + {z}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}{z} \]

      9. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\left|y\right| + \color{blue}{\sqrt{{z}^{2}}}\right)}{z} \]

      10. pow2 N/A

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

      11. rem-sqrt-square-rev N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

          \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{{y}^{2}}} + \left|z\right|\right)}{z} \]

      14. sqrt-pow1 N/A

          \[\leadsto \frac{x \cdot \left(\color{blue}{{y}^{\left(\frac{2}{2}\right)}} + \left|z\right|\right)}{z} \]

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. unpow1 N/A

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

      18. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left({y}^{\color{blue}{\left(\frac{2}{2}\right)}} + \left|z\right|\right)}{z} \]

      19. sqrt-pow1 N/A

          \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{y}^{2}}} + \left|z\right|\right)}{z} \]

      20. pow2 N/A

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

      21. sqrt-prod N/A

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

      22. rem-sqrt-square-rev N/A

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

      23. pow2 N/A

          \[\leadsto \frac{x \cdot \left(\sqrt{y} \cdot \sqrt{y} + \sqrt{\color{blue}{{z}^{2}}}\right)}{z} \]

      24. sqrt-pow1 N/A

          \[\leadsto \frac{x \cdot \left(\sqrt{y} \cdot \sqrt{y} + \color{blue}{{z}^{\left(\frac{2}{2}\right)}}\right)}{z} \]

      25. metadata-eval N/A

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

      26. unpow1 N/A

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

      27. lower-fma.f64 N/A

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

      28. lower-sqrt.f64 N/A

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

      29. lower-sqrt.f64 42.3

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

   4. Applied rewrites 42.3%

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

   5. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{z}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{z}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \frac{y \cdot {\left(\sqrt{-1}\right)}^{2}}{z}}\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y \cdot {\left(\sqrt{-1}\right)}^{2}}{z}} \]

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \frac{y \cdot {\left(\sqrt{-1}\right)}^{2}}{z} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot x\right) \cdot \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot y}}{z} \]

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. associate-/l* N/A

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

      16. associate-*l/ N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 49.1

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

   7. Applied rewrites 49.1%

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

   if 3.1999999999999999e-18 < x

   1. Initial program 73.0%

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

      1. lift-+.f64 N/A

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

      2. unpow1 N/A

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

      3. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left({y}^{\color{blue}{\left(\frac{2}{2}\right)}} + z\right)}{z} \]

      4. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{y}^{2}}} + z\right)}{z} \]

      5. pow2 N/A

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

      6. rem-sqrt-square-rev N/A

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

      7. unpow1 N/A

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

      8. metadata-eval N/A

         \[\leadsto \frac{x \cdot \left(\left|y\right| + {z}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}{z} \]

      9. sqrt-pow1 N/A

         \[\leadsto \frac{x \cdot \left(\left|y\right| + \color{blue}{\sqrt{{z}^{2}}}\right)}{z} \]

      10. pow2 N/A

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

      11. rem-sqrt-square-rev N/A

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

      12. rem-sqrt-square-rev N/A

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

      13. pow2 N/A

          \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{{y}^{2}}} + \left|z\right|\right)}{z} \]

      14. sqrt-pow1 N/A

          \[\leadsto \frac{x \cdot \left(\color{blue}{{y}^{\left(\frac{2}{2}\right)}} + \left|z\right|\right)}{z} \]

      15. metadata-eval N/A

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

      16. unpow1 N/A

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

      17. unpow1 N/A

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

      18. metadata-eval N/A

          \[\leadsto \frac{x \cdot \left({y}^{\color{blue}{\left(\frac{2}{2}\right)}} + \left|z\right|\right)}{z} \]

      19. sqrt-pow1 N/A

          \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{y}^{2}}} + \left|z\right|\right)}{z} \]

      20. pow2 N/A

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

      21. sqrt-prod N/A

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

      22. rem-sqrt-square-rev N/A

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

      23. pow2 N/A

          \[\leadsto \frac{x \cdot \left(\sqrt{y} \cdot \sqrt{y} + \sqrt{\color{blue}{{z}^{2}}}\right)}{z} \]

      24. sqrt-pow1 N/A

          \[\leadsto \frac{x \cdot \left(\sqrt{y} \cdot \sqrt{y} + \color{blue}{{z}^{\left(\frac{2}{2}\right)}}\right)}{z} \]

      25. metadata-eval N/A

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

      26. unpow1 N/A

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

      27. lower-fma.f64 N/A

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

      28. lower-sqrt.f64 N/A

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

      29. lower-sqrt.f64 39.0

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

   4. Applied rewrites 39.0%

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

   5. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{z}} \]
   6. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{z}\right)} \]

      2. associate-/l* N/A

         \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \frac{y \cdot {\left(\sqrt{-1}\right)}^{2}}{z}}\right) \]

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

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y \cdot {\left(\sqrt{-1}\right)}^{2}}{z}} \]

      4. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \frac{y \cdot {\left(\sqrt{-1}\right)}^{2}}{z} \]

      5. *-commutative N/A

         \[\leadsto \left(-1 \cdot x\right) \cdot \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot y}}{z} \]

      6. unpow2 N/A

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

      7. rem-square-sqrt N/A

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. distribute-rgt-neg-in N/A

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

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

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

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

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

      13. metadata-eval N/A

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

      14. *-lft-identity N/A

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

      15. associate-/l* N/A

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

      16. associate-*l/ N/A

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

      17. lower-*.f64 N/A

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

      18. lower-/.f64 52.0

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

   7. Applied rewrites 52.0%

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

   9. Applied rewrites 59.8%

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

4. Final simplification 51.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.2 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 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 82.0%

   \[\frac{x \cdot \left(y + z\right)}{z} \]
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. associate-/l* N/A

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

   2. +-commutative N/A

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

   3. div-add N/A

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

   4. distribute-rgt-in N/A

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

   5. *-inverses N/A

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

   6. metadata-eval N/A

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

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

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

   8. *-commutative N/A

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

   9. associate-/l* N/A

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

   10. remove-double-neg N/A

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

   11. mul-1-neg N/A

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

   12. distribute-neg-in N/A

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

   13. distribute-lft-out N/A

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

   14. mul-1-neg N/A

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

   15. remove-double-neg N/A

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

   16. +-commutative N/A

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

   17. associate-/l* N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f64 N/A

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

   20. lower-/.f64 95.1

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

5. Applied rewrites 95.1%

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

6. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-/.f64 93.2

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

8. Applied rewrites 93.2%

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

Alternative 5: 48.1% accurate, 1.2× speedup

Math

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

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 (/ y z))))

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 * (y / z));
}

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

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 * (y / z))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	return Float64(x_s * Float64(x_m * Float64(y / z)))
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 * (y / z));
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[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.0%

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

   1. lift-+.f64 N/A

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

   2. unpow1 N/A

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

   3. metadata-eval N/A

      \[\leadsto \frac{x \cdot \left({y}^{\color{blue}{\left(\frac{2}{2}\right)}} + z\right)}{z} \]

   4. sqrt-pow1 N/A

      \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{y}^{2}}} + z\right)}{z} \]

   5. pow2 N/A

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

   6. rem-sqrt-square-rev N/A

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

   7. unpow1 N/A

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

   8. metadata-eval N/A

      \[\leadsto \frac{x \cdot \left(\left|y\right| + {z}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}{z} \]

   9. sqrt-pow1 N/A

      \[\leadsto \frac{x \cdot \left(\left|y\right| + \color{blue}{\sqrt{{z}^{2}}}\right)}{z} \]

   10. pow2 N/A

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

   11. rem-sqrt-square-rev N/A

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

   12. rem-sqrt-square-rev N/A

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

   13. pow2 N/A

       \[\leadsto \frac{x \cdot \left(\sqrt{\color{blue}{{y}^{2}}} + \left|z\right|\right)}{z} \]

   14. sqrt-pow1 N/A

       \[\leadsto \frac{x \cdot \left(\color{blue}{{y}^{\left(\frac{2}{2}\right)}} + \left|z\right|\right)}{z} \]

   15. metadata-eval N/A

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

   16. unpow1 N/A

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

   17. unpow1 N/A

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

   18. metadata-eval N/A

       \[\leadsto \frac{x \cdot \left({y}^{\color{blue}{\left(\frac{2}{2}\right)}} + \left|z\right|\right)}{z} \]

   19. sqrt-pow1 N/A

       \[\leadsto \frac{x \cdot \left(\color{blue}{\sqrt{{y}^{2}}} + \left|z\right|\right)}{z} \]

   20. pow2 N/A

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

   21. sqrt-prod N/A

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

   22. rem-sqrt-square-rev N/A

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

   23. pow2 N/A

       \[\leadsto \frac{x \cdot \left(\sqrt{y} \cdot \sqrt{y} + \sqrt{\color{blue}{{z}^{2}}}\right)}{z} \]

   24. sqrt-pow1 N/A

       \[\leadsto \frac{x \cdot \left(\sqrt{y} \cdot \sqrt{y} + \color{blue}{{z}^{\left(\frac{2}{2}\right)}}\right)}{z} \]

   25. metadata-eval N/A

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

   26. unpow1 N/A

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

   27. lower-fma.f64 N/A

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

   28. lower-sqrt.f64 N/A

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

   29. lower-sqrt.f64 41.6

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

4. Applied rewrites 41.6%

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

5. Taylor expanded in y around -inf

   \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{z}} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{x \cdot \left(y \cdot {\left(\sqrt{-1}\right)}^{2}\right)}{z}\right)} \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{x \cdot \frac{y \cdot {\left(\sqrt{-1}\right)}^{2}}{z}}\right) \]

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

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(x\right)\right) \cdot \frac{y \cdot {\left(\sqrt{-1}\right)}^{2}}{z}} \]

   4. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(-1 \cdot x\right)} \cdot \frac{y \cdot {\left(\sqrt{-1}\right)}^{2}}{z} \]

   5. *-commutative N/A

      \[\leadsto \left(-1 \cdot x\right) \cdot \frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot y}}{z} \]

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. associate-*r/ N/A

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

   9. mul-1-neg N/A

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

   10. distribute-rgt-neg-in N/A

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

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

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

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

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

   13. metadata-eval N/A

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

   14. *-lft-identity N/A

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

   15. associate-/l* N/A

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

   16. associate-*l/ N/A

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

   17. lower-*.f64 N/A

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

   18. lower-/.f64 49.7

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

7. Applied rewrites 49.7%

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

9. Applied rewrites 49.7%

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

10. Final simplification 49.7%

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

Developer Target 1: 96.7% 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 2024359 
(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))