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

Percentage Accurate: 84.8% → 97.6%

Time: 7.5s

Alternatives: 8

Speedup: 0.6×

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

real(8) function code(x, y, z)
    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: 84.8% 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

real(8) function code(x, y, z)
    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: 97.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 4 \cdot 10^{+52}:\\ \;\;\;\;x\_m - \frac{x\_m \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m - x\_m \cdot \frac{z}{y}\\ \end{array} \end{array} \]

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 4e+52) (- x_m (/ (* x_m z) y)) (- x_m (* x_m (/ z y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 4e+52) {
		tmp = x_m - ((x_m * z) / y);
	} else {
		tmp = x_m - (x_m * (z / y));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    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 <= 4d+52) then
        tmp = x_m - ((x_m * z) / y)
    else
        tmp = x_m - (x_m * (z / y))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 4e+52) {
		tmp = x_m - ((x_m * z) / y);
	} else {
		tmp = x_m - (x_m * (z / y));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 4e+52:
		tmp = x_m - ((x_m * z) / y)
	else:
		tmp = x_m - (x_m * (z / y))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 4e+52)
		tmp = Float64(x_m - Float64(Float64(x_m * z) / y));
	else
		tmp = Float64(x_m - Float64(x_m * Float64(z / y)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 4e+52)
		tmp = x_m - ((x_m * z) / y);
	else
		tmp = x_m - (x_m * (z / y));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 4e+52], N[(x$95$m - N[(N[(x$95$m * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x$95$m - N[(x$95$m * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 4e52

   1. Initial program 86.2%

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

      1. remove-double-neg 86.2%

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

      2. distribute-frac-neg2 86.2%

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

      3. distribute-frac-neg 86.2%

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

      4. distribute-rgt-neg-in 86.2%

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

      5. associate-/l* 91.5%

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

      6. distribute-frac-neg 91.5%

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

      7. distribute-frac-neg2 91.5%

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

      8. remove-double-neg 91.5%

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

      9. div-sub 91.5%

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

      10. *-inverses 91.5%

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

   3. Simplified 91.5%

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

   5. Taylor expanded in z around 0 92.7%

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

      1. associate-*r/ 92.7%

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

      2. associate-*r* 92.7%

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

      3. mul-1-neg 92.7%

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

   7. Simplified 92.7%

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

   if 4e52 < x

   1. Initial program 80.1%

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

      1. remove-double-neg 80.1%

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

      2. distribute-frac-neg2 80.1%

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

      3. distribute-frac-neg 80.1%

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

      4. distribute-rgt-neg-in 80.1%

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

      5. associate-/l* 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. distribute-frac-neg2 99.9%

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

      8. remove-double-neg 99.9%

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

      9. div-sub 99.9%

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

      10. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 90.8%

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

      1. associate-*r/ 90.8%

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

      2. associate-*r* 90.8%

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

      3. mul-1-neg 90.8%

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

   7. Simplified 90.8%

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

      1. associate-/l* 99.9%

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

      2. add-sqr-sqrt 99.8%

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

      3. sqrt-unprod 68.2%

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

      4. sqr-neg 68.2%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 44.7%

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

      7. cancel-sign-sub-inv 44.7%

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

      8. associate-/l* 38.7%

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

      9. associate-/l* 44.7%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 68.2%

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

      12. sqr-neg 68.2%

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

      13. sqrt-unprod 99.8%

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

      14. add-sqr-sqrt 99.9%

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

   9. Applied egg-rr 99.9%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+52}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-21}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;y \leq 0.017:\\ \;\;\;\;z \cdot \frac{x\_m}{-y}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y -3.5e-21) x_m (if (<= y 0.017) (* 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) {
	double tmp;
	if (y <= -3.5e-21) {
		tmp = x_m;
	} else if (y <= 0.017) {
		tmp = z * (x_m / -y);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    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 (y <= (-3.5d-21)) then
        tmp = x_m
    else if (y <= 0.017d0) then
        tmp = z * (x_m / -y)
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -3.5e-21) {
		tmp = x_m;
	} else if (y <= 0.017) {
		tmp = z * (x_m / -y);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= -3.5e-21:
		tmp = x_m
	elif y <= 0.017:
		tmp = z * (x_m / -y)
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -3.5e-21)
		tmp = x_m;
	elseif (y <= 0.017)
		tmp = Float64(z * Float64(x_m / Float64(-y)));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= -3.5e-21)
		tmp = x_m;
	elseif (y <= 0.017)
		tmp = z * (x_m / -y);
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, -3.5e-21], x$95$m, If[LessEqual[y, 0.017], N[(z * N[(x$95$m / (-y)), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -3.5000000000000003e-21 or 0.017000000000000001 < y

   1. Initial program 76.4%

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

      1. remove-double-neg 76.4%

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

      2. distribute-frac-neg2 76.4%

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

      3. distribute-frac-neg 76.4%

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

      4. distribute-rgt-neg-in 76.4%

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

      5. associate-/l* 99.8%

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

      6. distribute-frac-neg 99.8%

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

      7. distribute-frac-neg2 99.8%

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

      8. remove-double-neg 99.8%

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

      9. div-sub 99.9%

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

      10. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 77.5%

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

   if -3.5000000000000003e-21 < y < 0.017000000000000001

   1. Initial program 92.9%

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

      1. remove-double-neg 92.9%

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

      2. distribute-frac-neg2 92.9%

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

      3. distribute-frac-neg 92.9%

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

      4. distribute-rgt-neg-in 92.9%

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

      5. associate-/l* 86.8%

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

      6. distribute-frac-neg 86.8%

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

      7. distribute-frac-neg2 86.8%

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

      8. remove-double-neg 86.8%

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

      9. div-sub 86.8%

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

      10. *-inverses 86.8%

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

   3. Simplified 86.8%

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

   5. Taylor expanded in z around inf 73.1%

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

      1. mul-1-neg 73.1%

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

      2. distribute-frac-neg2 73.1%

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

      3. *-commutative 73.1%

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

      4. associate-/l* 74.5%

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

   7. Simplified 74.5%

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

Alternative 3: 71.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-28}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;y \leq 0.085:\\ \;\;\;\;x\_m \cdot \frac{-z}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (* x_s (if (<= y -4.6e-28) x_m (if (<= y 0.085) (* x_m (/ (- z) y)) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -4.6e-28) {
		tmp = x_m;
	} else if (y <= 0.085) {
		tmp = x_m * (-z / y);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    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 (y <= (-4.6d-28)) then
        tmp = x_m
    else if (y <= 0.085d0) then
        tmp = x_m * (-z / y)
    else
        tmp = x_m
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -4.6e-28) {
		tmp = x_m;
	} else if (y <= 0.085) {
		tmp = x_m * (-z / y);
	} else {
		tmp = x_m;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if y <= -4.6e-28:
		tmp = x_m
	elif y <= 0.085:
		tmp = x_m * (-z / y)
	else:
		tmp = x_m
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (y <= -4.6e-28)
		tmp = x_m;
	elseif (y <= 0.085)
		tmp = Float64(x_m * Float64(Float64(-z) / y));
	else
		tmp = x_m;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (y <= -4.6e-28)
		tmp = x_m;
	elseif (y <= 0.085)
		tmp = x_m * (-z / y);
	else
		tmp = x_m;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[y, -4.6e-28], x$95$m, If[LessEqual[y, 0.085], N[(x$95$m * N[((-z) / y), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -4.59999999999999971e-28 or 0.0850000000000000061 < y

   1. Initial program 77.2%

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

      1. remove-double-neg 77.2%

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

      2. distribute-frac-neg2 77.2%

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

      3. distribute-frac-neg 77.2%

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

      4. distribute-rgt-neg-in 77.2%

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

      5. associate-/l* 99.1%

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

      6. distribute-frac-neg 99.1%

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

      7. distribute-frac-neg2 99.1%

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

      8. remove-double-neg 99.1%

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

      9. div-sub 99.1%

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

      10. *-inverses 99.1%

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

   3. Simplified 99.1%

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

   5. Taylor expanded in z around 0 76.6%

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

   if -4.59999999999999971e-28 < y < 0.0850000000000000061

   1. Initial program 92.7%

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

      1. remove-double-neg 92.7%

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

      2. distribute-frac-neg2 92.7%

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

      3. distribute-frac-neg 92.7%

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

      4. distribute-rgt-neg-in 92.7%

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

      5. associate-/l* 87.2%

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

      6. distribute-frac-neg 87.2%

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

      7. distribute-frac-neg2 87.2%

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

      8. remove-double-neg 87.2%

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

      9. div-sub 87.2%

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

      10. *-inverses 87.2%

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

   3. Simplified 87.2%

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

   5. Taylor expanded in z around inf 73.7%

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

      1. mul-1-neg 73.7%

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

      2. distribute-frac-neg2 73.7%

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

      3. associate-*r/ 64.7%

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

   7. Simplified 64.7%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{-28}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 0.085:\\ \;\;\;\;x \cdot \frac{-z}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.5 \cdot 10^{-34}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m - x\_m \cdot \frac{z}{y}\\ \end{array} \end{array} \]

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 2.5e-34) (/ (* x_m (- y z)) y) (- x_m (* x_m (/ z y))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.5e-34) {
		tmp = (x_m * (y - z)) / y;
	} else {
		tmp = x_m - (x_m * (z / y));
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    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 <= 2.5d-34) then
        tmp = (x_m * (y - z)) / y
    else
        tmp = x_m - (x_m * (z / y))
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 2.5e-34) {
		tmp = (x_m * (y - z)) / y;
	} else {
		tmp = x_m - (x_m * (z / y));
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	tmp = 0
	if x_m <= 2.5e-34:
		tmp = (x_m * (y - z)) / y
	else:
		tmp = x_m - (x_m * (z / y))
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 2.5e-34)
		tmp = Float64(Float64(x_m * Float64(y - z)) / y);
	else
		tmp = Float64(x_m - Float64(x_m * Float64(z / y)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (x_m <= 2.5e-34)
		tmp = (x_m * (y - z)) / y;
	else
		tmp = x_m - (x_m * (z / y));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 2.5e-34], N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(x$95$m - N[(x$95$m * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.5000000000000001e-34

   1. Initial program 85.3%

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

   if 2.5000000000000001e-34 < x

   1. Initial program 85.1%

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

      1. remove-double-neg 85.1%

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

      2. distribute-frac-neg2 85.1%

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

      3. distribute-frac-neg 85.1%

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

      4. distribute-rgt-neg-in 85.1%

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

      5. associate-/l* 99.8%

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

      6. distribute-frac-neg 99.8%

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

      7. distribute-frac-neg2 99.8%

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

      8. remove-double-neg 99.8%

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

      9. div-sub 99.9%

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

      10. *-inverses 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in z around 0 93.9%

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

      1. associate-*r/ 93.9%

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

      2. associate-*r* 93.9%

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

      3. mul-1-neg 93.9%

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

   7. Simplified 93.9%

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

      1. associate-/l* 99.9%

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

      2. add-sqr-sqrt 99.8%

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

      3. sqrt-unprod 78.7%

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

      4. sqr-neg 78.7%

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

      5. sqrt-unprod 0.0%

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

      6. add-sqr-sqrt 51.4%

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

      7. cancel-sign-sub-inv 51.4%

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

      8. associate-/l* 47.4%

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

      9. associate-/l* 51.4%

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

      10. add-sqr-sqrt 0.0%

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

      11. sqrt-unprod 78.7%

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

      12. sqr-neg 78.7%

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

      13. sqrt-unprod 99.8%

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

      14. add-sqr-sqrt 99.9%

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

   9. Applied egg-rr 99.9%

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

Alternative 5: 94.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{+116}:\\ \;\;\;\;x\_m - x\_m \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x\_m}{-y}\\ \end{array} \end{array} \]

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 (<= z 2.9e+116) (- x_m (* x_m (/ z y))) (* z (/ x_m (- 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 (z <= 2.9e+116) {
		tmp = x_m - (x_m * (z / y));
	} else {
		tmp = z * (x_m / -y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2.9d+116) then
        tmp = x_m - (x_m * (z / y))
    else
        tmp = z * (x_m / -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 (z <= 2.9e+116) {
		tmp = x_m - (x_m * (z / y));
	} else {
		tmp = z * (x_m / -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 z <= 2.9e+116:
		tmp = x_m - (x_m * (z / y))
	else:
		tmp = z * (x_m / -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 (z <= 2.9e+116)
		tmp = Float64(x_m - Float64(x_m * Float64(z / y)));
	else
		tmp = Float64(z * Float64(x_m / Float64(-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 (z <= 2.9e+116)
		tmp = x_m - (x_m * (z / y));
	else
		tmp = z * (x_m / -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[z, 2.9e+116], N[(x$95$m - N[(x$95$m * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(x$95$m / (-y)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.9000000000000001e116

   1. Initial program 84.8%

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

      1. remove-double-neg 84.8%

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

      2. distribute-frac-neg2 84.8%

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

      3. distribute-frac-neg 84.8%

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

      4. distribute-rgt-neg-in 84.8%

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

      5. associate-/l* 95.8%

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

      6. distribute-frac-neg 95.8%

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

      7. distribute-frac-neg2 95.8%

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

      8. remove-double-neg 95.8%

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

      9. div-sub 95.8%

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

      10. *-inverses 95.8%

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

   3. Simplified 95.8%

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

   5. Taylor expanded in z around 0 93.1%

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

      1. associate-*r/ 93.1%

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

      2. associate-*r* 93.1%

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

      3. mul-1-neg 93.1%

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

   7. Simplified 93.1%

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

      1. associate-/l* 95.9%

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

      2. add-sqr-sqrt 40.6%

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

      3. sqrt-unprod 55.8%

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

      4. sqr-neg 55.8%

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

      5. sqrt-unprod 31.4%

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

      6. add-sqr-sqrt 56.0%

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

      7. cancel-sign-sub-inv 56.0%

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

      8. associate-/l* 52.5%

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

      9. associate-/l* 56.0%

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

      10. add-sqr-sqrt 31.4%

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

      11. sqrt-unprod 55.8%

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

      12. sqr-neg 55.8%

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

      13. sqrt-unprod 40.6%

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

      14. add-sqr-sqrt 95.9%

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

   9. Applied egg-rr 95.9%

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

   if 2.9000000000000001e116 < z

   1. Initial program 87.0%

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

      1. remove-double-neg 87.0%

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

      2. distribute-frac-neg2 87.0%

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

      3. distribute-frac-neg 87.0%

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

      4. distribute-rgt-neg-in 87.0%

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

      5. associate-/l* 78.7%

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

      6. distribute-frac-neg 78.7%

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

      7. distribute-frac-neg2 78.7%

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

      8. remove-double-neg 78.7%

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

      9. div-sub 78.7%

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

      10. *-inverses 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in z around inf 85.1%

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

      1. mul-1-neg 85.1%

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

      2. distribute-frac-neg2 85.1%

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

      3. *-commutative 85.1%

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

      4. associate-/l* 89.4%

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

   7. Simplified 89.4%

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

Alternative 6: 94.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2.9 \cdot 10^{+116}:\\ \;\;\;\;x\_m \cdot \left(1 - \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x\_m}{-y}\\ \end{array} \end{array} \]

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 (<= z 2.9e+116) (* x_m (- 1.0 (/ z y))) (* z (/ x_m (- 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 (z <= 2.9e+116) {
		tmp = x_m * (1.0 - (z / y));
	} else {
		tmp = z * (x_m / -y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 2.9d+116) then
        tmp = x_m * (1.0d0 - (z / y))
    else
        tmp = z * (x_m / -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 (z <= 2.9e+116) {
		tmp = x_m * (1.0 - (z / y));
	} else {
		tmp = z * (x_m / -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 z <= 2.9e+116:
		tmp = x_m * (1.0 - (z / y))
	else:
		tmp = z * (x_m / -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 (z <= 2.9e+116)
		tmp = Float64(x_m * Float64(1.0 - Float64(z / y)));
	else
		tmp = Float64(z * Float64(x_m / Float64(-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 (z <= 2.9e+116)
		tmp = x_m * (1.0 - (z / y));
	else
		tmp = z * (x_m / -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[z, 2.9e+116], N[(x$95$m * N[(1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(z * N[(x$95$m / (-y)), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.9000000000000001e116

   1. Initial program 84.8%

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

      1. remove-double-neg 84.8%

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

      2. distribute-frac-neg2 84.8%

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

      3. distribute-frac-neg 84.8%

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

      4. distribute-rgt-neg-in 84.8%

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

      5. associate-/l* 95.8%

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

      6. distribute-frac-neg 95.8%

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

      7. distribute-frac-neg2 95.8%

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

      8. remove-double-neg 95.8%

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

      9. div-sub 95.8%

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

      10. *-inverses 95.8%

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

   3. Simplified 95.8%

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

   if 2.9000000000000001e116 < z

   1. Initial program 87.0%

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

      1. remove-double-neg 87.0%

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

      2. distribute-frac-neg2 87.0%

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

      3. distribute-frac-neg 87.0%

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

      4. distribute-rgt-neg-in 87.0%

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

      5. associate-/l* 78.7%

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

      6. distribute-frac-neg 78.7%

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

      7. distribute-frac-neg2 78.7%

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

      8. remove-double-neg 78.7%

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

      9. div-sub 78.7%

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

      10. *-inverses 78.7%

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

   3. Simplified 78.7%

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

   5. Taylor expanded in z around inf 85.1%

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

      1. mul-1-neg 85.1%

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

      2. distribute-frac-neg2 85.1%

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

      3. *-commutative 85.1%

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

      4. associate-/l* 89.4%

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

   7. Simplified 89.4%

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

Alternative 7: 51.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 6 \cdot 10^{+197}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{y}\\ \end{array} \end{array} \]

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 6e+197) x_m (* y (/ x_m 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 <= 6e+197) {
		tmp = x_m;
	} else {
		tmp = y * (x_m / y);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    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 <= 6d+197) then
        tmp = x_m
    else
        tmp = y * (x_m / 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 <= 6e+197) {
		tmp = x_m;
	} else {
		tmp = y * (x_m / 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 <= 6e+197:
		tmp = x_m
	else:
		tmp = y * (x_m / 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 (x_m <= 6e+197)
		tmp = x_m;
	else
		tmp = Float64(y * Float64(x_m / 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 <= 6e+197)
		tmp = x_m;
	else
		tmp = y * (x_m / 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[x$95$m, 6e+197], x$95$m, N[(y * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 6.0000000000000004e197

   1. Initial program 86.7%

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

      1. remove-double-neg 86.7%

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

      2. distribute-frac-neg2 86.7%

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

      3. distribute-frac-neg 86.7%

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

      4. distribute-rgt-neg-in 86.7%

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

      5. associate-/l* 92.3%

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

      6. distribute-frac-neg 92.3%

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

      7. distribute-frac-neg2 92.3%

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

      8. remove-double-neg 92.3%

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

      9. div-sub 92.3%

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

      10. *-inverses 92.3%

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

   3. Simplified 92.3%

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

   5. Taylor expanded in z around 0 50.6%

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

   if 6.0000000000000004e197 < x

   1. Initial program 66.3%

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

   3. Taylor expanded in y around inf 16.3%

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

      1. *-commutative 16.3%

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

      2. associate-/l* 73.2%

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

   5. Applied egg-rr 73.2%

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

Alternative 8: 50.7% accurate, 7.0× 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 = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z)
    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 85.2%

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

   1. remove-double-neg 85.2%

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

   2. distribute-frac-neg2 85.2%

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

   3. distribute-frac-neg 85.2%

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

   4. distribute-rgt-neg-in 85.2%

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

   5. associate-/l* 92.9%

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

   6. distribute-frac-neg 92.9%

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

   7. distribute-frac-neg2 92.9%

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

   8. remove-double-neg 92.9%

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

   9. div-sub 92.9%

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

   10. *-inverses 92.9%

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

3. Simplified 92.9%

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

5. Taylor expanded in z around 0 49.8%

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

Developer Target 1: 96.3% accurate, 0.4× 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

real(8) function code(x, y, z)
    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 2024157 
(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))