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

Percentage Accurate: 84.1% → 96.5%

Time: 8.5s

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: 84.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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: 96.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.2 \cdot 10^{-221}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-x\_m}{y}, z, x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \end{array} \]

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 5.2e-221) (fma (/ (- x_m) y) z x_m) (* 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 <= 5.2e-221) {
		tmp = fma((-x_m / y), z, x_m);
	} 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 <= 5.2e-221)
		tmp = fma(Float64(Float64(-x_m) / y), z, x_m);
	else
		tmp = Float64(x_m * Float64(1.0 - Float64(z / y)));
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[x$95$m, 5.2e-221], N[(N[((-x$95$m) / y), $MachinePrecision] * z + x$95$m), $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 < 5.2000000000000004e-221

   1. Initial program 87.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

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

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

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

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

      6. --lowering--.f64 93.4

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

   4. Applied egg-rr 93.4%

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

      1. clear-num N/A

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

      2. associate-/l/ N/A

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

      3. associate-/r/ N/A

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

      4. sub-neg N/A

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

      5. distribute-rgt-in N/A

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

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

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

      7. distribute-rgt-neg-out N/A

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

      8. distribute-rgt-in N/A

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

      9. div-inv N/A

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

      10. associate-*r* N/A

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

      11. *-commutative N/A

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

      12. associate-*r* N/A

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

      13. div-inv N/A

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

      14. *-inverses N/A

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

      15. *-lft-identity N/A

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

      16. +-commutative N/A

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

      17. frac-2neg N/A

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

      18. distribute-rgt-neg-out N/A

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

      19. remove-double-neg N/A

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

      20. associate-*r/ N/A

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

      21. *-commutative N/A

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

      22. accelerator-lowering-fma.f64 N/A

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

   6. Applied egg-rr 95.2%

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

   if 5.2000000000000004e-221 < x

   1. Initial program 80.8%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

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

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

      7. /-lowering-/.f64 98.2

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

   4. Applied egg-rr 98.2%

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

4. Final simplification 96.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.2 \cdot 10^{-221}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-x}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x\_m \cdot \left(y - z\right)}{y} \leq 10^{+298}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x\_m}{y}\\ \end{array} \end{array} \]

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 (- y z)) y) 1e+298) 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 * (y - z)) / y) <= 1e+298) {
		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 * (y - z)) / y) <= 1d+298) 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 * (y - z)) / y) <= 1e+298) {
		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 * (y - z)) / y) <= 1e+298:
		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 (Float64(Float64(x_m * Float64(y - z)) / y) <= 1e+298)
		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 * (y - z)) / y) <= 1e+298)
		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[N[(N[(x$95$m * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], 1e+298], x$95$m, N[(y * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < 9.9999999999999996e297

   1. Initial program 89.3%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 45.6%

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

   if 9.9999999999999996e297 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 58.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. --lowering--.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 46.1%

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

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq 10^{+298}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := -z \cdot \frac{x\_m}{y}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -850000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+50}:\\ \;\;\;\;x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (- (* z (/ x_m y)))))
   (* x_s (if (<= z -850000000.0) t_0 (if (<= z 1.35e+50) x_m t_0)))))

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 = -(z * (x_m / y));
	double tmp;
	if (z <= -850000000.0) {
		tmp = t_0;
	} else if (z <= 1.35e+50) {
		tmp = x_m;
	} else {
		tmp = t_0;
	}
	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) :: tmp
    t_0 = -(z * (x_m / y))
    if (z <= (-850000000.0d0)) then
        tmp = t_0
    else if (z <= 1.35d+50) then
        tmp = x_m
    else
        tmp = t_0
    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 = -(z * (x_m / y));
	double tmp;
	if (z <= -850000000.0) {
		tmp = t_0;
	} else if (z <= 1.35e+50) {
		tmp = x_m;
	} else {
		tmp = t_0;
	}
	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 = -(z * (x_m / y))
	tmp = 0
	if z <= -850000000.0:
		tmp = t_0
	elif z <= 1.35e+50:
		tmp = x_m
	else:
		tmp = t_0
	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(z * Float64(x_m / y)))
	tmp = 0.0
	if (z <= -850000000.0)
		tmp = t_0;
	elseif (z <= 1.35e+50)
		tmp = x_m;
	else
		tmp = t_0;
	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 = -(z * (x_m / y));
	tmp = 0.0;
	if (z <= -850000000.0)
		tmp = t_0;
	elseif (z <= 1.35e+50)
		tmp = x_m;
	else
		tmp = t_0;
	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[(z * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision])}, N[(x$95$s * If[LessEqual[z, -850000000.0], t$95$0, If[LessEqual[z, 1.35e+50], x$95$m, t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5e8 or 1.35e50 < z

   1. Initial program 89.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-*r/ N/A

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

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

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. neg-lowering-neg.f64 78.1

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

   5. Simplified 78.1%

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

      1. frac-2neg N/A

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

      2. distribute-rgt-neg-out N/A

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

      3. remove-double-neg N/A

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

      4. associate-*r/ N/A

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

      5. *-commutative N/A

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

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

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

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

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

      8. neg-lowering-neg.f64 79.8

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

   7. Applied egg-rr 79.8%

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

   if -8.5e8 < z < 1.35e50

   1. Initial program 80.0%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 73.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -850000000:\\ \;\;\;\;-z \cdot \frac{x}{y}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+50}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-z \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.3 \cdot 10^{-83}:\\ \;\;\;\;x\_m - \frac{x\_m \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \end{array} \]

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 1.3e-83) (- x_m (/ (* x_m z) y)) (* 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 <= 1.3e-83) {
		tmp = x_m - ((x_m * z) / y);
	} 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 <= 1.3d-83) then
        tmp = x_m - ((x_m * z) / y)
    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 <= 1.3e-83) {
		tmp = x_m - ((x_m * z) / y);
	} 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 <= 1.3e-83:
		tmp = x_m - ((x_m * z) / y)
	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 <= 1.3e-83)
		tmp = Float64(x_m - Float64(Float64(x_m * z) / y));
	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 <= 1.3e-83)
		tmp = x_m - ((x_m * z) / y);
	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, 1.3e-83], N[(x$95$m - N[(N[(x$95$m * z), $MachinePrecision] / y), $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 < 1.30000000000000004e-83

   1. Initial program 89.1%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

      4. distribute-lft-out-- N/A

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

      5. *-rgt-identity N/A

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

      6. associate-/l* N/A

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

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

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

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

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

      9. *-lowering-*.f64 95.3

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

   5. Simplified 95.3%

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

   if 1.30000000000000004e-83 < x

   1. Initial program 76.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

      4. div-sub N/A

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

      5. *-inverses N/A

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

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

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

      7. /-lowering-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{-83}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 - \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 96.1% 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 84.7%

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

   1. associate-/l* N/A

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

   2. *-commutative N/A

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

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

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

   4. div-sub N/A

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

   5. *-inverses N/A

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

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

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

   7. /-lowering-/.f64 95.4

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

4. Applied egg-rr 95.4%

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

5. Final simplification 95.4%

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

Alternative 6: 51.5% accurate, 20.0× speedup

Math

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

FPCore

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

C

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

Fortran

x\_m = 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 84.7%

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

3. Taylor expanded in y around inf

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

5. Simplified 44.6%

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

Developer Target 1: 95.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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 2024199 
(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))