Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3

Percentage Accurate: 85.1% → 96.1%

Time: 6.8s

Alternatives: 7

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5 \cdot 10^{-75}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{y} \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= x_m 1.5e-75) (/ (* x_m (- y z)) y) (* (/ (- y z) y) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.5e-75) {
		tmp = (x_m * (y - z)) / y;
	} else {
		tmp = ((y - z) / y) * x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x_m <= 1.5d-75) then
        tmp = (x_m * (y - z)) / y
    else
        tmp = ((y - z) / y) * x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 1.5e-75) {
		tmp = (x_m * (y - z)) / y;
	} else {
		tmp = ((y - z) / y) * x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 1.5e-75:
		tmp = (x_m * (y - z)) / y
	else:
		tmp = ((y - 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)
	tmp = 0.0
	if (x_m <= 1.5e-75)
		tmp = Float64(Float64(x_m * Float64(y - z)) / y);
	else
		tmp = Float64(Float64(Float64(y - z) / y) * x_m);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 1.5e-75)
		tmp = (x_m * (y - z)) / y;
	else
		tmp = ((y - z) / y) * x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 1.5e-75], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / y), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.4999999999999999e-75

   1. Initial program 83.5%

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

   if 1.4999999999999999e-75 < x

   1. Initial program 79.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

Alternative 2: 78.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \left(-z\right)}{y}\\ t_1 := \frac{x\_m \cdot \left(y - z\right)}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+304}:\\ \;\;\;\;\frac{-x\_m}{y} \cdot z\\ \mathbf{elif}\;t\_1 \leq -4 \cdot 10^{-236}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+226}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;t\_1 \leq 10^{+301}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y} \cdot y\\ \end{array} \end{array} \end{array} \]

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 (- z)) y)) (t_1 (/ (* x_m (- y z)) y)))
   (*
    x_s
    (if (<= t_1 -5e+304)
      (* (/ (- x_m) y) z)
      (if (<= t_1 -4e-236)
        t_0
        (if (<= t_1 2e+226) x_m (if (<= t_1 1e+301) t_0 (* (/ x_m y) 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 * -z) / y;
	double t_1 = (x_m * (y - z)) / y;
	double tmp;
	if (t_1 <= -5e+304) {
		tmp = (-x_m / y) * z;
	} else if (t_1 <= -4e-236) {
		tmp = t_0;
	} else if (t_1 <= 2e+226) {
		tmp = x_m;
	} else if (t_1 <= 1e+301) {
		tmp = t_0;
	} else {
		tmp = (x_m / y) * 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) :: t_1
    real(8) :: tmp
    t_0 = (x_m * -z) / y
    t_1 = (x_m * (y - z)) / y
    if (t_1 <= (-5d+304)) then
        tmp = (-x_m / y) * z
    else if (t_1 <= (-4d-236)) then
        tmp = t_0
    else if (t_1 <= 2d+226) then
        tmp = x_m
    else if (t_1 <= 1d+301) then
        tmp = t_0
    else
        tmp = (x_m / y) * 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 * -z) / y;
	double t_1 = (x_m * (y - z)) / y;
	double tmp;
	if (t_1 <= -5e+304) {
		tmp = (-x_m / y) * z;
	} else if (t_1 <= -4e-236) {
		tmp = t_0;
	} else if (t_1 <= 2e+226) {
		tmp = x_m;
	} else if (t_1 <= 1e+301) {
		tmp = t_0;
	} else {
		tmp = (x_m / y) * 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 * -z) / y
	t_1 = (x_m * (y - z)) / y
	tmp = 0
	if t_1 <= -5e+304:
		tmp = (-x_m / y) * z
	elif t_1 <= -4e-236:
		tmp = t_0
	elif t_1 <= 2e+226:
		tmp = x_m
	elif t_1 <= 1e+301:
		tmp = t_0
	else:
		tmp = (x_m / y) * 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(-z)) / y)
	t_1 = Float64(Float64(x_m * Float64(y - z)) / y)
	tmp = 0.0
	if (t_1 <= -5e+304)
		tmp = Float64(Float64(Float64(-x_m) / y) * z);
	elseif (t_1 <= -4e-236)
		tmp = t_0;
	elseif (t_1 <= 2e+226)
		tmp = x_m;
	elseif (t_1 <= 1e+301)
		tmp = t_0;
	else
		tmp = Float64(Float64(x_m / y) * 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 * -z) / y;
	t_1 = (x_m * (y - z)) / y;
	tmp = 0.0;
	if (t_1 <= -5e+304)
		tmp = (-x_m / y) * z;
	elseif (t_1 <= -4e-236)
		tmp = t_0;
	elseif (t_1 <= 2e+226)
		tmp = x_m;
	elseif (t_1 <= 1e+301)
		tmp = t_0;
	else
		tmp = (x_m / y) * 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 * (-z)), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, -5e+304], N[(N[((-x$95$m) / y), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[t$95$1, -4e-236], t$95$0, If[LessEqual[t$95$1, 2e+226], x$95$m, If[LessEqual[t$95$1, 1e+301], t$95$0, N[(N[(x$95$m / y), $MachinePrecision] * y), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < -4.9999999999999997e304

   1. Initial program 47.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{y}} \]
   4. Step-by-step derivation

   5. Applied rewrites 53.7%

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

   if -4.9999999999999997e304 < (/.f64 (*.f64 x (-.f64 y z)) y) < -4.0000000000000002e-236 or 1.99999999999999992e226 < (/.f64 (*.f64 x (-.f64 y z)) y) < 1.00000000000000005e301

   1. Initial program 98.9%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 55.3%

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

   if -4.0000000000000002e-236 < (/.f64 (*.f64 x (-.f64 y z)) y) < 1.99999999999999992e226

   1. Initial program 95.3%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 54.0%

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

   if 1.00000000000000005e301 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 57.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 46.6%

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

Alternative 3: 75.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \left(y - z\right)}{y}\\ t_1 := \frac{-x\_m}{y} \cdot z\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq -4 \cdot 10^{-236}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 2 \cdot 10^{+226}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;t\_0 \leq 10^{+301}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y} \cdot y\\ \end{array} \end{array} \end{array} \]

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)) y)) (t_1 (* (/ (- x_m) y) z)))
   (*
    x_s
    (if (<= t_0 -4e-236)
      t_1
      (if (<= t_0 2e+226) x_m (if (<= t_0 1e+301) t_1 (* (/ x_m y) 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)) / y;
	double t_1 = (-x_m / y) * z;
	double tmp;
	if (t_0 <= -4e-236) {
		tmp = t_1;
	} else if (t_0 <= 2e+226) {
		tmp = x_m;
	} else if (t_0 <= 1e+301) {
		tmp = t_1;
	} else {
		tmp = (x_m / y) * 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) :: t_1
    real(8) :: tmp
    t_0 = (x_m * (y - z)) / y
    t_1 = (-x_m / y) * z
    if (t_0 <= (-4d-236)) then
        tmp = t_1
    else if (t_0 <= 2d+226) then
        tmp = x_m
    else if (t_0 <= 1d+301) then
        tmp = t_1
    else
        tmp = (x_m / y) * 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)) / y;
	double t_1 = (-x_m / y) * z;
	double tmp;
	if (t_0 <= -4e-236) {
		tmp = t_1;
	} else if (t_0 <= 2e+226) {
		tmp = x_m;
	} else if (t_0 <= 1e+301) {
		tmp = t_1;
	} else {
		tmp = (x_m / y) * 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)) / y
	t_1 = (-x_m / y) * z
	tmp = 0
	if t_0 <= -4e-236:
		tmp = t_1
	elif t_0 <= 2e+226:
		tmp = x_m
	elif t_0 <= 1e+301:
		tmp = t_1
	else:
		tmp = (x_m / y) * 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)) / y)
	t_1 = Float64(Float64(Float64(-x_m) / y) * z)
	tmp = 0.0
	if (t_0 <= -4e-236)
		tmp = t_1;
	elseif (t_0 <= 2e+226)
		tmp = x_m;
	elseif (t_0 <= 1e+301)
		tmp = t_1;
	else
		tmp = Float64(Float64(x_m / y) * 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)) / y;
	t_1 = (-x_m / y) * z;
	tmp = 0.0;
	if (t_0 <= -4e-236)
		tmp = t_1;
	elseif (t_0 <= 2e+226)
		tmp = x_m;
	elseif (t_0 <= 1e+301)
		tmp = t_1;
	else
		tmp = (x_m / y) * 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] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[((-x$95$m) / y), $MachinePrecision] * z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, -4e-236], t$95$1, If[LessEqual[t$95$0, 2e+226], x$95$m, If[LessEqual[t$95$0, 1e+301], t$95$1, N[(N[(x$95$m / y), $MachinePrecision] * y), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < -4.0000000000000002e-236 or 1.99999999999999992e226 < (/.f64 (*.f64 x (-.f64 y z)) y) < 1.00000000000000005e301

   1. Initial program 81.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{y}} \]
   4. Step-by-step derivation

   5. Applied rewrites 51.0%

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

   if -4.0000000000000002e-236 < (/.f64 (*.f64 x (-.f64 y z)) y) < 1.99999999999999992e226

   1. Initial program 95.3%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 54.0%

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

   if 1.00000000000000005e301 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 57.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 46.6%

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

Alternative 4: 53.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \left(y - z\right)}{y} \leq 5 \cdot 10^{+252}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{y} \cdot y\\ \end{array} \end{array} \]

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)) y) 5e+252) x_m (* (/ x_m y) 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)) / y) <= 5e+252) {
		tmp = x_m;
	} else {
		tmp = (x_m / y) * 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)) / y) <= 5d+252) then
        tmp = x_m
    else
        tmp = (x_m / y) * 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)) / y) <= 5e+252) {
		tmp = x_m;
	} else {
		tmp = (x_m / y) * 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)) / y) <= 5e+252:
		tmp = x_m
	else:
		tmp = (x_m / y) * 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)) / y) <= 5e+252)
		tmp = x_m;
	else
		tmp = Float64(Float64(x_m / y) * 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)) / y) <= 5e+252)
		tmp = x_m;
	else
		tmp = (x_m / y) * 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] / y), $MachinePrecision], 5e+252], x$95$m, N[(N[(x$95$m / y), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < 4.9999999999999997e252

   1. Initial program 86.5%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 50.6%

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

   if 4.9999999999999997e252 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 61.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 42.9%

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

Alternative 5: 97.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{x\_m}{y}, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{y} \cdot x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= x_m 5e-6) (fma (- z) (/ x_m y) x_m) (* (/ (- y z) y) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5e-6) {
		tmp = fma(-z, (x_m / y), x_m);
	} else {
		tmp = ((y - 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)
	tmp = 0.0
	if (x_m <= 5e-6)
		tmp = fma(Float64(-z), Float64(x_m / y), x_m);
	else
		tmp = Float64(Float64(Float64(y - 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_] := N[(x$95$s * If[LessEqual[x$95$m, 5e-6], N[((-z) * N[(x$95$m / y), $MachinePrecision] + x$95$m), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / y), $MachinePrecision] * x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.00000000000000041e-6

   1. Initial program 84.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lift--.f64 N/A

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

      6. flip-- N/A

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

      7. frac-2neg N/A

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

      8. frac-times N/A

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

      9. lower-/.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. difference-of-squares N/A

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

      12. lift--.f64 N/A

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

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

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

      14. lower-*.f64 N/A

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

      15. +-commutative N/A

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

      16. lower-+.f64 N/A

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

      17. lower-neg.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lower-neg.f64 N/A

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

      20. +-commutative N/A

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

      21. lower-+.f64 47.7

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

   4. Applied rewrites 47.7%

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

   5. Taylor expanded in y around inf

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

   7. Applied rewrites 93.5%

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

   if 5.00000000000000041e-6 < x

   1. Initial program 76.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

4. Final simplification 95.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(-z, \frac{x}{y}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.8% accurate, 1.0× speedup

Math

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

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 (- z) (/ x_m 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(-z, (x_m / 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(-z), Float64(x_m / 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[((-z) * N[(x$95$m / y), $MachinePrecision] + x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 82.0%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. lift--.f64 N/A

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

   6. flip-- N/A

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

   7. frac-2neg N/A

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

   8. frac-times N/A

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

   9. lower-/.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. difference-of-squares N/A

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

   12. lift--.f64 N/A

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

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

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

   14. lower-*.f64 N/A

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

   15. +-commutative N/A

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

   16. lower-+.f64 N/A

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

   17. lower-neg.f64 N/A

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

   18. lower-*.f64 N/A

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

   19. lower-neg.f64 N/A

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

   20. +-commutative N/A

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

   21. lower-+.f64 46.2

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

4. Applied rewrites 46.2%

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

5. Taylor expanded in y around inf

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

7. Applied rewrites 92.4%

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

8. Final simplification 92.4%

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

Alternative 7: 50.4% accurate, 20.0× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m =     private
x\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x_s, x_m, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x_s * x_m
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.0%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 47.4%

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

Developer Target 1: 95.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z < -2.060202331921739 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z < 1.6939766013828526 \cdot 10^{+213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (< z -2.060202331921739e+104)
   (- x (/ (* z x) y))
   (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z < -2.060202331921739e+104) {
		tmp = x - ((z * x) / y);
	} else if (z < 1.6939766013828526e+213) {
		tmp = x / (y / (y - z));
	} else {
		tmp = (y - z) * (x / y);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z < (-2.060202331921739d+104)) then
        tmp = x - ((z * x) / y)
    else if (z < 1.6939766013828526d+213) then
        tmp = x / (y / (y - z))
    else
        tmp = (y - z) * (x / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z < -2.060202331921739e+104) {
		tmp = x - ((z * x) / y);
	} else if (z < 1.6939766013828526e+213) {
		tmp = x / (y / (y - z));
	} else {
		tmp = (y - z) * (x / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z < -2.060202331921739e+104:
		tmp = x - ((z * x) / y)
	elif z < 1.6939766013828526e+213:
		tmp = x / (y / (y - z))
	else:
		tmp = (y - z) * (x / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z < -2.060202331921739e+104)
		tmp = Float64(x - Float64(Float64(z * x) / y));
	elseif (z < 1.6939766013828526e+213)
		tmp = Float64(x / Float64(y / Float64(y - z)));
	else
		tmp = Float64(Float64(y - z) * Float64(x / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z < -2.060202331921739e+104)
		tmp = x - ((z * x) / y);
	elseif (z < 1.6939766013828526e+213)
		tmp = x / (y / (y - z));
	else
		tmp = (y - z) * (x / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Less[z, -2.060202331921739e+104], N[(x - N[(N[(z * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[Less[z, 1.6939766013828526e+213], N[(x / N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(x / y), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2025021 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -206020233192173900000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- x (/ (* z x) y)) (if (< z 1693976601382852600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ x (/ y (- y z))) (* (- y z) (/ x y)))))

  (/ (* x (- y z)) y))