Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3

Percentage Accurate: 88.1% → 99.5%

Time: 9.9s

Alternatives: 11

Speedup: 0.9×

• 20.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))

C

double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

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) + 1.0d0)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

Python

def code(x, y, z):
	return (x * ((y - z) + 1.0)) / z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(Float64(y - z) + 1.0)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * ((y - z) + 1.0)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* x (+ (- y z) 1.0)) z))

C

double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

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) + 1.0d0)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (x * ((y - z) + 1.0)) / z;
}

Python

def code(x, y, z):
	return (x * ((y - z) + 1.0)) / z

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(Float64(y - z) + 1.0)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * ((y - z) + 1.0)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 5.8 \cdot 10^{+80}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, y, x\_m\right)}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(-1 + \frac{y + 1}{z}\right)\\ \end{array} \end{array} \]

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 5.8e+80)
    (- (/ (fma x_m y x_m) z) x_m)
    (* x_m (+ -1.0 (/ (+ y 1.0) z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z) {
	double tmp;
	if (x_m <= 5.8e+80) {
		tmp = (fma(x_m, y, x_m) / z) - x_m;
	} else {
		tmp = x_m * (-1.0 + ((y + 1.0) / z));
	}
	return x_s * tmp;
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z)
	tmp = 0.0
	if (x_m <= 5.8e+80)
		tmp = Float64(Float64(fma(x_m, y, x_m) / z) - x_m);
	else
		tmp = Float64(x_m * Float64(-1.0 + Float64(Float64(y + 1.0) / z)));
	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, 5.8e+80], N[(N[(N[(x$95$m * y + x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision], N[(x$95$m * N[(-1.0 + N[(N[(y + 1.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 5.79999999999999971e80

   1. Initial program 92.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

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

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

      5. associate-/l* N/A

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

      6. *-rgt-identity N/A

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

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

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

   5. Simplified 98.0%

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

   if 5.79999999999999971e80 < x

   1. Initial program 66.9%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0

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

      1. sub-neg N/A

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

      2. *-rgt-identity N/A

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

      3. associate-*r/ N/A

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

      4. distribute-rgt1-in N/A

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

      5. +-commutative N/A

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

      6. metadata-eval N/A

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

      7. distribute-neg-frac N/A

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

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

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

      9. associate-/l* N/A

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

      10. associate-*l/ N/A

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

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

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

      12. metadata-eval N/A

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

      13. *-rgt-identity N/A

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

      14. metadata-eval N/A

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

      15. +-commutative N/A

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

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

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

      17. /-lowering-/.f64 N/A

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

      18. +-lowering-+.f64 99.9

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

   7. Simplified 99.9%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5.8 \cdot 10^{+80}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, y, x\right)}{z} - x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-1 + \frac{y + 1}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{y}{z}, x\_m, -x\_m\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.5 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-7}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, y, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (fma (/ y z) x_m (- x_m))))
   (*
    x_s
    (if (<= z -4.5e+17) t_0 (if (<= z 1.15e-7) (/ (fma x_m y x_m) z) 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 = fma((y / z), x_m, -x_m);
	double tmp;
	if (z <= -4.5e+17) {
		tmp = t_0;
	} else if (z <= 1.15e-7) {
		tmp = fma(x_m, y, x_m) / z;
	} 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 = fma(Float64(y / z), x_m, Float64(-x_m))
	tmp = 0.0
	if (z <= -4.5e+17)
		tmp = t_0;
	elseif (z <= 1.15e-7)
		tmp = Float64(fma(x_m, y, x_m) / z);
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(y / z), $MachinePrecision] * x$95$m + (-x$95$m)), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -4.5e+17], t$95$0, If[LessEqual[z, 1.15e-7], N[(N[(x$95$m * y + x$95$m), $MachinePrecision] / z), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.5e17 or 1.14999999999999997e-7 < z

   1. Initial program 75.2%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

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

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

      5. associate-/l* N/A

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

      6. *-rgt-identity N/A

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

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

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

   5. Simplified 93.4%

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

   6. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 92.6

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

   8. Simplified 92.6%

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

      1. sub-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. neg-lowering-neg.f64 99.2

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

   10. Applied egg-rr 99.2%

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

   if -4.5e17 < z < 1.14999999999999997e-7

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 99.9

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

   5. Simplified 99.9%

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

Alternative 3: 94.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot y}{z} - x\_m\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7.5 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, y, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (- (/ (* x_m y) z) x_m)))
   (*
    x_s
    (if (<= z -7.5e+17) t_0 (if (<= z 5.4e-19) (/ (fma x_m y x_m) z) 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 = ((x_m * y) / z) - x_m;
	double tmp;
	if (z <= -7.5e+17) {
		tmp = t_0;
	} else if (z <= 5.4e-19) {
		tmp = fma(x_m, y, x_m) / z;
	} 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(Float64(x_m * y) / z) - x_m)
	tmp = 0.0
	if (z <= -7.5e+17)
		tmp = t_0;
	elseif (z <= 5.4e-19)
		tmp = Float64(fma(x_m, y, x_m) / z);
	else
		tmp = t_0;
	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_] := Block[{t$95$0 = N[(N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -7.5e+17], t$95$0, If[LessEqual[z, 5.4e-19], N[(N[(x$95$m * y + x$95$m), $MachinePrecision] / z), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.5e17 or 5.4000000000000002e-19 < z

   1. Initial program 75.6%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

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

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

      5. associate-/l* N/A

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

      6. *-rgt-identity N/A

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

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

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

   5. Simplified 93.5%

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

   6. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 92.7

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

   8. Simplified 92.7%

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

   if -7.5e17 < z < 5.4000000000000002e-19

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 99.8

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

   5. Simplified 99.8%

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

Alternative 4: 87.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.9 \cdot 10^{+57}:\\ \;\;\;\;-x\_m\\ \mathbf{elif}\;z \leq 0.0012:\\ \;\;\;\;\left(y + 1\right) \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

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 (<= z -5.9e+57)
    (- x_m)
    (if (<= z 0.0012) (* (+ y 1.0) (/ x_m z)) (- (/ x_m z) 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 (z <= -5.9e+57) {
		tmp = -x_m;
	} else if (z <= 0.0012) {
		tmp = (y + 1.0) * (x_m / z);
	} else {
		tmp = (x_m / z) - 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 (z <= (-5.9d+57)) then
        tmp = -x_m
    else if (z <= 0.0012d0) then
        tmp = (y + 1.0d0) * (x_m / z)
    else
        tmp = (x_m / z) - 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 (z <= -5.9e+57) {
		tmp = -x_m;
	} else if (z <= 0.0012) {
		tmp = (y + 1.0) * (x_m / z);
	} else {
		tmp = (x_m / z) - 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 z <= -5.9e+57:
		tmp = -x_m
	elif z <= 0.0012:
		tmp = (y + 1.0) * (x_m / z)
	else:
		tmp = (x_m / z) - 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 (z <= -5.9e+57)
		tmp = Float64(-x_m);
	elseif (z <= 0.0012)
		tmp = Float64(Float64(y + 1.0) * Float64(x_m / z));
	else
		tmp = Float64(Float64(x_m / z) - 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 (z <= -5.9e+57)
		tmp = -x_m;
	elseif (z <= 0.0012)
		tmp = (y + 1.0) * (x_m / z);
	else
		tmp = (x_m / z) - 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[z, -5.9e+57], (-x$95$m), If[LessEqual[z, 0.0012], N[(N[(y + 1.0), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -5.90000000000000013e57

   1. Initial program 70.3%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. neg-lowering-neg.f64 87.2

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

   5. Simplified 87.2%

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

   if -5.90000000000000013e57 < z < 0.00119999999999999989

   1. Initial program 99.2%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 97.2%

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

   if 0.00119999999999999989 < z

   1. Initial program 77.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 82.4

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

   5. Simplified 82.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.9 \cdot 10^{+57}:\\ \;\;\;\;-x\\ \mathbf{elif}\;z \leq 0.0012:\\ \;\;\;\;\left(y + 1\right) \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{+57}:\\ \;\;\;\;-x\_m\\ \mathbf{elif}\;z \leq 1.05:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, y, x\_m\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \end{array} \end{array} \]

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 (<= z -1.5e+57)
    (- x_m)
    (if (<= z 1.05) (/ (fma x_m y x_m) z) (- (/ x_m z) 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 (z <= -1.5e+57) {
		tmp = -x_m;
	} else if (z <= 1.05) {
		tmp = fma(x_m, y, x_m) / z;
	} else {
		tmp = (x_m / z) - 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 (z <= -1.5e+57)
		tmp = Float64(-x_m);
	elseif (z <= 1.05)
		tmp = Float64(fma(x_m, y, x_m) / z);
	else
		tmp = Float64(Float64(x_m / z) - x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -1.5e+57], (-x$95$m), If[LessEqual[z, 1.05], N[(N[(x$95$m * y + x$95$m), $MachinePrecision] / z), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -1.5e57

   1. Initial program 70.3%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. neg-lowering-neg.f64 87.2

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

   5. Simplified 87.2%

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

   if -1.5e57 < z < 1.05000000000000004

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. accelerator-lowering-fma.f64 97.2

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

   5. Simplified 97.2%

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

   if 1.05000000000000004 < z

   1. Initial program 77.7%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 82.4

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

   5. Simplified 82.4%

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

Alternative 6: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := \frac{x\_m \cdot y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1200:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.62 \cdot 10^{+118}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (/ (* x_m y) z)))
   (*
    x_s
    (if (<= y -1200.0) t_0 (if (<= y 1.62e+118) (- (/ x_m z) 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 = (x_m * y) / z;
	double tmp;
	if (y <= -1200.0) {
		tmp = t_0;
	} else if (y <= 1.62e+118) {
		tmp = (x_m / z) - 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 = (x_m * y) / z
    if (y <= (-1200.0d0)) then
        tmp = t_0
    else if (y <= 1.62d+118) then
        tmp = (x_m / z) - 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 = (x_m * y) / z;
	double tmp;
	if (y <= -1200.0) {
		tmp = t_0;
	} else if (y <= 1.62e+118) {
		tmp = (x_m / z) - 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 = (x_m * y) / z
	tmp = 0
	if y <= -1200.0:
		tmp = t_0
	elif y <= 1.62e+118:
		tmp = (x_m / z) - 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(x_m * y) / z)
	tmp = 0.0
	if (y <= -1200.0)
		tmp = t_0;
	elseif (y <= 1.62e+118)
		tmp = Float64(Float64(x_m / z) - 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 = (x_m * y) / z;
	tmp = 0.0;
	if (y <= -1200.0)
		tmp = t_0;
	elseif (y <= 1.62e+118)
		tmp = (x_m / z) - 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[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1200.0], t$95$0, If[LessEqual[y, 1.62e+118], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1200 or 1.6199999999999999e118 < y

   1. Initial program 93.5%

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

   3. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 81.5

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

   5. Simplified 81.5%

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

   if -1200 < y < 1.6199999999999999e118

   1. Initial program 83.3%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 95.7

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

   5. Simplified 95.7%

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

Alternative 7: 85.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := y \cdot \frac{x\_m}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -130000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{x\_m}{z} - x\_m\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \end{array} \]

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 (* y (/ x_m z))))
   (*
    x_s
    (if (<= y -130000.0) t_0 (if (<= y 9.6e+117) (- (/ x_m z) 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 = y * (x_m / z);
	double tmp;
	if (y <= -130000.0) {
		tmp = t_0;
	} else if (y <= 9.6e+117) {
		tmp = (x_m / z) - 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 = y * (x_m / z)
    if (y <= (-130000.0d0)) then
        tmp = t_0
    else if (y <= 9.6d+117) then
        tmp = (x_m / z) - 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 = y * (x_m / z);
	double tmp;
	if (y <= -130000.0) {
		tmp = t_0;
	} else if (y <= 9.6e+117) {
		tmp = (x_m / z) - 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 = y * (x_m / z)
	tmp = 0
	if y <= -130000.0:
		tmp = t_0
	elif y <= 9.6e+117:
		tmp = (x_m / z) - 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(y * Float64(x_m / z))
	tmp = 0.0
	if (y <= -130000.0)
		tmp = t_0;
	elseif (y <= 9.6e+117)
		tmp = Float64(Float64(x_m / z) - 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 = y * (x_m / z);
	tmp = 0.0;
	if (y <= -130000.0)
		tmp = t_0;
	elseif (y <= 9.6e+117)
		tmp = (x_m / z) - 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[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -130000.0], t$95$0, If[LessEqual[y, 9.6e+117], N[(N[(x$95$m / z), $MachinePrecision] - x$95$m), $MachinePrecision], t$95$0]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e5 or 9.5999999999999996e117 < y

   1. Initial program 93.5%

      \[\frac{x \cdot \left(\left(y - z\right) + 1\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

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

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

      7. --lowering--.f64 87.9

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

   4. Applied egg-rr 87.9%

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

   5. Taylor expanded in y around inf

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

   7. Simplified 78.3%

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

   if -1.3e5 < y < 9.5999999999999996e117

   1. Initial program 83.3%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 95.7

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

   5. Simplified 95.7%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -130000:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+117}:\\ \;\;\;\;\frac{x}{z} - x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 97.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, x\_m, -x\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m, y, x\_m\right)}{z} - x\_m\\ \end{array} \end{array} \]

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 (<= z -5e+17) (fma (/ y z) x_m (- x_m)) (- (/ (fma x_m y x_m) z) 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 (z <= -5e+17) {
		tmp = fma((y / z), x_m, -x_m);
	} else {
		tmp = (fma(x_m, y, x_m) / z) - 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 (z <= -5e+17)
		tmp = fma(Float64(y / z), x_m, Float64(-x_m));
	else
		tmp = Float64(Float64(fma(x_m, y, x_m) / z) - x_m);
	end
	return Float64(x_s * tmp)
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_] := N[(x$95$s * If[LessEqual[z, -5e+17], N[(N[(y / z), $MachinePrecision] * x$95$m + (-x$95$m)), $MachinePrecision], N[(N[(N[(x$95$m * y + x$95$m), $MachinePrecision] / z), $MachinePrecision] - x$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -5e17

   1. Initial program 72.8%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

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

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

      5. associate-/l* N/A

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

      6. *-rgt-identity N/A

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

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

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

   5. Simplified 91.8%

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

   6. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 91.8

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

   8. Simplified 91.8%

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

      1. sub-neg N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

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

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

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

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

      6. neg-lowering-neg.f64 100.0

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

   10. Applied egg-rr 100.0%

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

   if -5e17 < z

   1. Initial program 92.9%

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

   3. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-inverses N/A

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

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

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

      5. associate-/l* N/A

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

      6. *-rgt-identity N/A

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

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

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

   5. Simplified 98.4%

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

Alternative 9: 65.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;-x\_m\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;-x\_m\\ \end{array} \end{array} \]

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 (<= z -1.0) (- x_m) (if (<= z 5.4e-19) (/ x_m z) (- 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 (z <= -1.0) {
		tmp = -x_m;
	} else if (z <= 5.4e-19) {
		tmp = x_m / z;
	} 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 (z <= (-1.0d0)) then
        tmp = -x_m
    else if (z <= 5.4d-19) then
        tmp = x_m / z
    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 (z <= -1.0) {
		tmp = -x_m;
	} else if (z <= 5.4e-19) {
		tmp = x_m / z;
	} 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 z <= -1.0:
		tmp = -x_m
	elif z <= 5.4e-19:
		tmp = x_m / z
	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 (z <= -1.0)
		tmp = Float64(-x_m);
	elseif (z <= 5.4e-19)
		tmp = Float64(x_m / z);
	else
		tmp = Float64(-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 (z <= -1.0)
		tmp = -x_m;
	elseif (z <= 5.4e-19)
		tmp = x_m / z;
	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[z, -1.0], (-x$95$m), If[LessEqual[z, 5.4e-19], N[(x$95$m / z), $MachinePrecision], (-x$95$m)]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 5.4000000000000002e-19 < z

   1. Initial program 76.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

      2. neg-lowering-neg.f64 78.4

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

   5. Simplified 78.4%

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

   if -1 < z < 5.4000000000000002e-19

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. sub-neg N/A

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

      4. *-inverses N/A

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

      5. metadata-eval N/A

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

      6. distribute-lft-in N/A

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

      7. associate-/l* N/A

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

      8. *-rgt-identity N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. unsub-neg N/A

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

      12. --lowering--.f64 N/A

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

      13. /-lowering-/.f64 53.3

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

   5. Simplified 53.3%

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

   6. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 53.3

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

   8. Simplified 53.3%

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

Alternative 10: 66.5% accurate, 1.5× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (- (/ x_m z) 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 / z) - 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 / z) - 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 / z) - 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 / z) - 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 * Float64(Float64(x_m / z) - 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 / z) - x_m);
end

Wolfram

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

Derivation

1. Initial program 87.5%

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

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. div-sub N/A

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

   3. sub-neg N/A

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

   4. *-inverses N/A

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

   5. metadata-eval N/A

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

   6. distribute-lft-in N/A

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

   7. associate-/l* N/A

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

   8. *-rgt-identity N/A

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

   9. *-commutative N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. --lowering--.f64 N/A

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

   13. /-lowering-/.f64 66.7

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

5. Simplified 66.7%

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

Alternative 11: 39.5% accurate, 7.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z) :precision binary64 (* x_s (- 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 * Float64(-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 87.5%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. neg-lowering-neg.f64 42.2

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

5. Simplified 42.2%

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

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(1 + y\right) \cdot \frac{x}{z} - x\\ \mathbf{if}\;x < -2.71483106713436 \cdot 10^{-162}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x < 3.874108816439546 \cdot 10^{-197}:\\ \;\;\;\;\left(x \cdot \left(\left(y - z\right) + 1\right)\right) \cdot \frac{1}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (- (* (+ 1.0 y) (/ x z)) x)))
   (if (< x -2.71483106713436e-162)
     t_0
     (if (< x 3.874108816439546e-197)
       (* (* x (+ (- y z) 1.0)) (/ 1.0 z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = ((1.0d0 + y) * (x / z)) - x
    if (x < (-2.71483106713436d-162)) then
        tmp = t_0
    else if (x < 3.874108816439546d-197) then
        tmp = (x * ((y - z) + 1.0d0)) * (1.0d0 / z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((1.0 + y) * (x / z)) - x;
	double tmp;
	if (x < -2.71483106713436e-162) {
		tmp = t_0;
	} else if (x < 3.874108816439546e-197) {
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((1.0 + y) * (x / z)) - x
	tmp = 0
	if x < -2.71483106713436e-162:
		tmp = t_0
	elif x < 3.874108816439546e-197:
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(1.0 + y) * Float64(x / z)) - x)
	tmp = 0.0
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = Float64(Float64(x * Float64(Float64(y - z) + 1.0)) * Float64(1.0 / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((1.0 + y) * (x / z)) - x;
	tmp = 0.0;
	if (x < -2.71483106713436e-162)
		tmp = t_0;
	elseif (x < 3.874108816439546e-197)
		tmp = (x * ((y - z) + 1.0)) * (1.0 / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(1.0 + y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision]}, If[Less[x, -2.71483106713436e-162], t$95$0, If[Less[x, 3.874108816439546e-197], N[(N[(x * N[(N[(y - z), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z)
  :name "Diagrams.TwoD.Segment.Bernstein:evaluateBernstein from diagrams-lib-1.3.0.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -67870776678359/25000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (* (+ 1 y) (/ x z)) x) (if (< x 1937054408219773/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (* x (+ (- y z) 1)) (/ 1 z)) (- (* (+ 1 y) (/ x z)) x))))

  (/ (* x (+ (- y z) 1.0)) z))