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

Percentage Accurate: 84.5% → 97.6%

Time: 9.0s

Alternatives: 6

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

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: 84.5% 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, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot \left(y - z\right)}{y}\\ t_1 := x\_m \cdot \left(1 - \frac{z}{y}\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 2 \cdot 10^{+103}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (let* ((t_0 (/ (* x_m (- y z)) y)) (t_1 (* x_m (- 1.0 (/ z y)))))
   (* x_s (if (<= t_0 2e+103) t_1 (if (<= t_0 5e+303) t_0 t_1)))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (y - z)) / y;
	double t_1 = x_m * (1.0 - (z / y));
	double tmp;
	if (t_0 <= 2e+103) {
		tmp = t_1;
	} else if (t_0 <= 5e+303) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (x_m * (y - z)) / y
    t_1 = x_m * (1.0d0 - (z / y))
    if (t_0 <= 2d+103) then
        tmp = t_1
    else if (t_0 <= 5d+303) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z) {
	double t_0 = (x_m * (y - z)) / y;
	double t_1 = x_m * (1.0 - (z / y));
	double tmp;
	if (t_0 <= 2e+103) {
		tmp = t_1;
	} else if (t_0 <= 5e+303) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z):
	t_0 = (x_m * (y - z)) / y
	t_1 = x_m * (1.0 - (z / y))
	tmp = 0
	if t_0 <= 2e+103:
		tmp = t_1
	elif t_0 <= 5e+303:
		tmp = t_0
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	t_0 = Float64(Float64(x_m * Float64(y - z)) / y)
	t_1 = Float64(x_m * Float64(1.0 - Float64(z / y)))
	tmp = 0.0
	if (t_0 <= 2e+103)
		tmp = t_1;
	elseif (t_0 <= 5e+303)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	t_0 = (x_m * (y - z)) / y;
	t_1 = x_m * (1.0 - (z / y));
	tmp = 0.0;
	if (t_0 <= 2e+103)
		tmp = t_1;
	elseif (t_0 <= 5e+303)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := Block[{t$95$0 = N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(x$95$m * N[(1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$0, 2e+103], t$95$1, If[LessEqual[t$95$0, 5e+303], t$95$0, t$95$1]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < 2e103 or 4.9999999999999997e303 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 82.8%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y - z}{y}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{y} - \color{blue}{\frac{z}{y}}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{y}{y}\right), \color{blue}{\left(\frac{z}{y}\right)}\right)\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\frac{\color{blue}{z}}{y}\right)\right)\right) \]

      6. /-lowering-/.f64 96.3%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{y}\right)\right)\right) \]

   3. Simplified 96.3%

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

   if 2e103 < (/.f64 (*.f64 x (-.f64 y z)) y) < 4.9999999999999997e303

   1. Initial program 99.5%

      \[\frac{x \cdot \left(y - z\right)}{y} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 72.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-23}:\\ \;\;\;\;0 - \frac{z}{\frac{y}{x\_m}}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+59}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - x\_m \cdot z}{y}\\ \end{array} \end{array} \]

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 (<= z -2.3e-23)
    (- 0.0 (/ z (/ y x_m)))
    (if (<= z 6.8e+59) x_m (/ (- 0.0 (* 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 (z <= -2.3e-23) {
		tmp = 0.0 - (z / (y / x_m));
	} else if (z <= 6.8e+59) {
		tmp = x_m;
	} else {
		tmp = (0.0 - (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 (z <= (-2.3d-23)) then
        tmp = 0.0d0 - (z / (y / x_m))
    else if (z <= 6.8d+59) then
        tmp = x_m
    else
        tmp = (0.0d0 - (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 (z <= -2.3e-23) {
		tmp = 0.0 - (z / (y / x_m));
	} else if (z <= 6.8e+59) {
		tmp = x_m;
	} else {
		tmp = (0.0 - (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 z <= -2.3e-23:
		tmp = 0.0 - (z / (y / x_m))
	elif z <= 6.8e+59:
		tmp = x_m
	else:
		tmp = (0.0 - (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 (z <= -2.3e-23)
		tmp = Float64(0.0 - Float64(z / Float64(y / x_m)));
	elseif (z <= 6.8e+59)
		tmp = x_m;
	else
		tmp = Float64(Float64(0.0 - Float64(x_m * z)) / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z)
	tmp = 0.0;
	if (z <= -2.3e-23)
		tmp = 0.0 - (z / (y / x_m));
	elseif (z <= 6.8e+59)
		tmp = x_m;
	else
		tmp = (0.0 - (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[z, -2.3e-23], N[(0.0 - N[(z / N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.8e+59], x$95$m, N[(N[(0.0 - N[(x$95$m * z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -2.3000000000000001e-23

   1. Initial program 87.9%

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

      1. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{y \cdot y - z \cdot z}{y + z}\right), y\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{\frac{y + z}{y \cdot y - z \cdot z}}\right), y\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\frac{y + z}{y \cdot y - z \cdot z}}\right), y\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y + z}{y \cdot y - z \cdot z}\right)\right), y\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{\frac{y \cdot y - z \cdot z}{y + z}}\right)\right), y\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y - z}\right)\right), y\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \left(y - z\right)\right)\right), y\right) \]

      8. --lowering--.f64 87.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, z\right)\right)\right), y\right) \]

   4. Applied egg-rr 87.8%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{1} \cdot \left(y - z\right)\right), y\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{1}{x}} \cdot \left(y - z\right)\right), y\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{\frac{1}{x}}{y - z}}\right), y\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y - z}{\frac{1}{x}}\right), y\right) \]

      5. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\frac{1}{x}\right)}\right), y\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(y - z\right)\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(y + \left(\mathsf{neg}\left(z\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(\left(\mathsf{neg}\left(z\right)\right) + y\right)\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - y\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(z - y\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, y\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, y\right), \left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), y\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, y\right), \left(\frac{-1}{x}\right)\right), y\right) \]

      16. /-lowering-/.f64 87.9%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, y\right), \mathsf{/.f64}\left(-1, x\right)\right), y\right) \]

   6. Applied egg-rr 87.9%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{z}, \mathsf{/.f64}\left(-1, x\right)\right), y\right) \]
   8. Step-by-step derivation

   9. Simplified 71.7%

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

       1. associate-/r/ N/A

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

       2. associate-/l* N/A

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

       3. times-frac N/A

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

       4. neg-mul-1 N/A

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

       5. clear-num N/A

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

       6. distribute-neg-frac N/A

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

       7. associate-/l/ N/A

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

       8. distribute-frac-neg2 N/A

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

       9. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{\frac{y}{x}}{z}}\right)\right) \]

       10. clear-num N/A

           \[\leadsto \mathsf{neg.f64}\left(\left(\frac{z}{\frac{y}{x}}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(z, \left(\frac{y}{x}\right)\right)\right) \]

       12. /-lowering-/.f64 73.3%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(y, x\right)\right)\right) \]

   11. Applied egg-rr 73.3%

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

   if -2.3000000000000001e-23 < z < 6.80000000000000012e59

   1. Initial program 78.9%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y - z}{y}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{y} - \color{blue}{\frac{z}{y}}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{y}{y}\right), \color{blue}{\left(\frac{z}{y}\right)}\right)\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\frac{\color{blue}{z}}{y}\right)\right)\right) \]

      6. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{y}\right)\right)\right) \]

   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

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

   7. Simplified 79.4%

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

   if 6.80000000000000012e59 < z

   1. Initial program 93.9%

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

      1. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{y \cdot y - z \cdot z}{y + z}\right), y\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{\frac{y + z}{y \cdot y - z \cdot z}}\right), y\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\frac{y + z}{y \cdot y - z \cdot z}}\right), y\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y + z}{y \cdot y - z \cdot z}\right)\right), y\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{\frac{y \cdot y - z \cdot z}{y + z}}\right)\right), y\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y - z}\right)\right), y\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \left(y - z\right)\right)\right), y\right) \]

      8. --lowering--.f64 93.8%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, z\right)\right)\right), y\right) \]

   4. Applied egg-rr 93.8%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{1} \cdot \left(y - z\right)\right), y\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{1}{x}} \cdot \left(y - z\right)\right), y\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{\frac{1}{x}}{y - z}}\right), y\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y - z}{\frac{1}{x}}\right), y\right) \]

      5. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\frac{1}{x}\right)}\right), y\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(y - z\right)\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(y + \left(\mathsf{neg}\left(z\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(\left(\mathsf{neg}\left(z\right)\right) + y\right)\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - y\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(z - y\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, y\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, y\right), \left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), y\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, y\right), \left(\frac{-1}{x}\right)\right), y\right) \]

      16. /-lowering-/.f64 93.8%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, y\right), \mathsf{/.f64}\left(-1, x\right)\right), y\right) \]

   6. Applied egg-rr 93.8%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{z}, \mathsf{/.f64}\left(-1, x\right)\right), y\right) \]
   8. Step-by-step derivation

   9. Simplified 82.2%

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

       1. frac-2neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(z\right)}{\mathsf{neg}\left(\frac{-1}{x}\right)}\right), y\right) \]

       2. distribute-frac-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{z}{\mathsf{neg}\left(\frac{-1}{x}\right)}\right)\right), y\right) \]

       3. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{z}{\frac{\mathsf{neg}\left(-1\right)}{x}}\right)\right), y\right) \]

       4. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(\frac{z}{\frac{1}{x}}\right)\right), y\right) \]

       5. un-div-inv N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(z \cdot \frac{1}{\frac{1}{x}}\right)\right), y\right) \]

       6. remove-double-div N/A

          \[\leadsto \mathsf{/.f64}\left(\left(\mathsf{neg}\left(z \cdot x\right)\right), y\right) \]

       7. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(z \cdot x\right)\right), y\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\left(x \cdot z\right)\right), y\right) \]

       9. *-lowering-*.f64 82.3%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(x, z\right)\right), y\right) \]

   11. Applied egg-rr 82.3%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-23}:\\ \;\;\;\;0 - \frac{z}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 6.8 \cdot 10^{+59}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{0 - x \cdot z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 72.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+52}:\\ \;\;\;\;x\_m\\ \mathbf{elif}\;y \leq 1800000000000:\\ \;\;\;\;0 - \frac{z}{\frac{y}{x\_m}}\\ \mathbf{else}:\\ \;\;\;\;x\_m\\ \end{array} \end{array} \]

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 -7.5e+52)
    x_m
    (if (<= y 1800000000000.0) (- 0.0 (/ z (/ y x_m))) x_m))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (y <= -7.5e+52) {
		tmp = x_m;
	} else if (y <= 1800000000000.0) {
		tmp = 0.0 - (z / (y / x_m));
	} 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 <= (-7.5d+52)) then
        tmp = x_m
    else if (y <= 1800000000000.0d0) then
        tmp = 0.0d0 - (z / (y / x_m))
    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 <= -7.5e+52) {
		tmp = x_m;
	} else if (y <= 1800000000000.0) {
		tmp = 0.0 - (z / (y / x_m));
	} 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 <= -7.5e+52:
		tmp = x_m
	elif y <= 1800000000000.0:
		tmp = 0.0 - (z / (y / x_m))
	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 <= -7.5e+52)
		tmp = x_m;
	elseif (y <= 1800000000000.0)
		tmp = Float64(0.0 - Float64(z / Float64(y / x_m)));
	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 <= -7.5e+52)
		tmp = x_m;
	elseif (y <= 1800000000000.0)
		tmp = 0.0 - (z / (y / x_m));
	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, -7.5e+52], x$95$m, If[LessEqual[y, 1800000000000.0], N[(0.0 - N[(z / N[(y / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], x$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -7.49999999999999995e52 or 1.8e12 < y

   1. Initial program 77.2%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y - z}{y}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{y} - \color{blue}{\frac{z}{y}}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{y}{y}\right), \color{blue}{\left(\frac{z}{y}\right)}\right)\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\frac{\color{blue}{z}}{y}\right)\right)\right) \]

      6. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{y}\right)\right)\right) \]

   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

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

   7. Simplified 83.0%

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

   if -7.49999999999999995e52 < y < 1.8e12

   1. Initial program 90.6%

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

      1. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{y \cdot y - z \cdot z}{y + z}\right), y\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(x \cdot \frac{1}{\frac{y + z}{y \cdot y - z \cdot z}}\right), y\right) \]

      3. un-div-inv N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{\frac{y + z}{y \cdot y - z \cdot z}}\right), y\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{y + z}{y \cdot y - z \cdot z}\right)\right), y\right) \]

      5. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{\frac{y \cdot y - z \cdot z}{y + z}}\right)\right), y\right) \]

      6. flip-- N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \left(\frac{1}{y - z}\right)\right), y\right) \]

      7. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \left(y - z\right)\right)\right), y\right) \]

      8. --lowering--.f64 90.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{\_.f64}\left(y, z\right)\right)\right), y\right) \]

   4. Applied egg-rr 90.5%

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

      1. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{x}{1} \cdot \left(y - z\right)\right), y\right) \]

      2. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{1}{x}} \cdot \left(y - z\right)\right), y\right) \]

      3. associate-/r/ N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{1}{\frac{\frac{1}{x}}{y - z}}\right), y\right) \]

      4. clear-num N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{y - z}{\frac{1}{x}}\right), y\right) \]

      5. frac-2neg N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\frac{1}{x}\right)}\right), y\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      7. neg-sub0 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(y - z\right)\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      8. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(y + \left(\mathsf{neg}\left(z\right)\right)\right)\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      9. +-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(0 - \left(\left(\mathsf{neg}\left(z\right)\right) + y\right)\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      10. associate--r+ N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - y\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      11. neg-sub0 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      12. remove-double-neg N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(z - y\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, y\right), \left(\mathsf{neg}\left(\frac{1}{x}\right)\right)\right), y\right) \]

      14. distribute-neg-frac N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, y\right), \left(\frac{\mathsf{neg}\left(1\right)}{x}\right)\right), y\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, y\right), \left(\frac{-1}{x}\right)\right), y\right) \]

      16. /-lowering-/.f64 90.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, y\right), \mathsf{/.f64}\left(-1, x\right)\right), y\right) \]

   6. Applied egg-rr 90.5%

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

   7. Taylor expanded in z around inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\color{blue}{z}, \mathsf{/.f64}\left(-1, x\right)\right), y\right) \]
   8. Step-by-step derivation

   9. Simplified 73.2%

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

       1. associate-/r/ N/A

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

       2. associate-/l* N/A

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

       3. times-frac N/A

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

       4. neg-mul-1 N/A

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

       5. clear-num N/A

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

       6. distribute-neg-frac N/A

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

       7. associate-/l/ N/A

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

       8. distribute-frac-neg2 N/A

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

       9. neg-lowering-neg.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\left(\frac{1}{\frac{\frac{y}{x}}{z}}\right)\right) \]

       10. clear-num N/A

           \[\leadsto \mathsf{neg.f64}\left(\left(\frac{z}{\frac{y}{x}}\right)\right) \]

       11. /-lowering-/.f64 N/A

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(z, \left(\frac{y}{x}\right)\right)\right) \]

       12. /-lowering-/.f64 73.5%

           \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(z, \mathsf{/.f64}\left(y, x\right)\right)\right) \]

   11. Applied egg-rr 73.5%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{+52}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1800000000000:\\ \;\;\;\;0 - \frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2 \cdot 10^{-31}:\\ \;\;\;\;x\_m + \frac{-1}{\frac{\frac{y}{x\_m}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z)
 :precision binary64
 (*
  x_s
  (if (<= x_m 1.2e-31)
    (+ x_m (/ -1.0 (/ (/ y x_m) z)))
    (* x_m (- 1.0 (/ 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 <= 1.2e-31) {
		tmp = x_m + (-1.0 / ((y / x_m) / z));
	} else {
		tmp = x_m * (1.0 - (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 <= 1.2d-31) then
        tmp = x_m + ((-1.0d0) / ((y / x_m) / z))
    else
        tmp = x_m * (1.0d0 - (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 <= 1.2e-31) {
		tmp = x_m + (-1.0 / ((y / x_m) / z));
	} else {
		tmp = x_m * (1.0 - (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 <= 1.2e-31:
		tmp = x_m + (-1.0 / ((y / x_m) / z))
	else:
		tmp = x_m * (1.0 - (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 <= 1.2e-31)
		tmp = Float64(x_m + Float64(-1.0 / Float64(Float64(y / x_m) / z)));
	else
		tmp = Float64(x_m * Float64(1.0 - 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 <= 1.2e-31)
		tmp = x_m + (-1.0 / ((y / x_m) / z));
	else
		tmp = x_m * (1.0 - (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, 1.2e-31], N[(x$95$m + N[(-1.0 / N[(N[(y / x$95$m), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 1.2e-31

   1. Initial program 85.2%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y - z}{y}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{y} - \color{blue}{\frac{z}{y}}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{y}{y}\right), \color{blue}{\left(\frac{z}{y}\right)}\right)\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\frac{\color{blue}{z}}{y}\right)\right)\right) \]

      6. /-lowering-/.f64 93.5%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{y}\right)\right)\right) \]

   3. Simplified 93.5%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-in N/A

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

      4. *-rgt-identity N/A

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

      5. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{z}{y}\right)\right)\right), \color{blue}{x}\right) \]

      6. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(\mathsf{neg}\left(\frac{1}{\frac{y}{z}}\right)\right)\right), x\right) \]

      7. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \frac{1}{\mathsf{neg}\left(\frac{y}{z}\right)}\right), x\right) \]

      8. un-div-inv N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{x}{\mathsf{neg}\left(\frac{y}{z}\right)}\right), x\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \left(\mathsf{neg}\left(\frac{y}{z}\right)\right)\right), x\right) \]

      10. neg-sub0 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \left(0 - \frac{y}{z}\right)\right), x\right) \]

      11. --lowering--.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(0, \left(\frac{y}{z}\right)\right)\right), x\right) \]

      12. /-lowering-/.f64 89.1%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(x, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(y, z\right)\right)\right), x\right) \]

   6. Applied egg-rr 89.1%

      \[\leadsto \color{blue}{\frac{x}{0 - \frac{y}{z}} + x} \]
   7. Step-by-step derivation

      1. clear-num N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{\frac{0 - \frac{y}{z}}{x}}\right), x\right) \]

      2. frac-2neg N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{\mathsf{neg}\left(1\right)}{\mathsf{neg}\left(\frac{0 - \frac{y}{z}}{x}\right)}\right), x\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{-1}{\mathsf{neg}\left(\frac{0 - \frac{y}{z}}{x}\right)}\right), x\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\frac{0 - \frac{y}{z}}{x}\right)\right)\right), x\right) \]

      5. sub0-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\frac{\mathsf{neg}\left(\frac{y}{z}\right)}{x}\right)\right)\right), x\right) \]

      6. distribute-frac-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{\frac{y}{z}}{x}\right)\right)\right)\right)\right), x\right) \]

      7. remove-double-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, \left(\frac{\frac{y}{z}}{x}\right)\right), x\right) \]

      8. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{y}{z}\right), x\right)\right), x\right) \]

      9. /-lowering-/.f64 93.4%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, z\right), x\right)\right), x\right) \]

   8. Applied egg-rr 93.4%

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

      1. associate-/l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, \left(\frac{y}{x \cdot z}\right)\right), x\right) \]

      2. associate-/r* N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, \left(\frac{\frac{y}{x}}{z}\right)\right), x\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\left(\frac{y}{x}\right), z\right)\right), x\right) \]

      4. /-lowering-/.f64 94.0%

         \[\leadsto \mathsf{+.f64}\left(\mathsf{/.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{/.f64}\left(y, x\right), z\right)\right), x\right) \]

   10. Applied egg-rr 94.0%

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

   if 1.2e-31 < x

   1. Initial program 81.8%

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

      1. associate-/l* N/A

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

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y - z}{y}\right)}\right) \]

      3. div-sub N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{y} - \color{blue}{\frac{z}{y}}\right)\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{y}{y}\right), \color{blue}{\left(\frac{z}{y}\right)}\right)\right) \]

      5. *-inverses N/A

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\frac{\color{blue}{z}}{y}\right)\right)\right) \]

      6. /-lowering-/.f64 99.9%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{y}\right)\right)\right) \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x \cdot \left(1 - \frac{z}{y}\right)} \]
   4. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.2 \cdot 10^{-31}:\\ \;\;\;\;x + \frac{-1}{\frac{\frac{y}{x}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 * (1.0d0 - (z / y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 84.3%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y - z}{y}\right)}\right) \]

   3. div-sub N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{y} - \color{blue}{\frac{z}{y}}\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{y}{y}\right), \color{blue}{\left(\frac{z}{y}\right)}\right)\right) \]

   5. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\frac{\color{blue}{z}}{y}\right)\right)\right) \]

   6. /-lowering-/.f64 95.2%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{y}\right)\right)\right) \]

3. Simplified 95.2%

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

Alternative 6: 50.8% 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 84.3%

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

   1. associate-/l* N/A

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

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(\frac{y - z}{y}\right)}\right) \]

   3. div-sub N/A

      \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{y}{y} - \color{blue}{\frac{z}{y}}\right)\right) \]

   4. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(\left(\frac{y}{y}\right), \color{blue}{\left(\frac{z}{y}\right)}\right)\right) \]

   5. *-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \left(\frac{\color{blue}{z}}{y}\right)\right)\right) \]

   6. /-lowering-/.f64 95.2%

      \[\leadsto \mathsf{*.f64}\left(x, \mathsf{\_.f64}\left(1, \mathsf{/.f64}\left(z, \color{blue}{y}\right)\right)\right) \]

3. Simplified 95.2%

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

5. Taylor expanded in z around 0

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

7. Simplified 52.1%

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

Developer Target 1: 95.7% 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 2024158 
(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))