Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B

Percentage Accurate: 99.8% → 99.9%

Time: 6.8s

Alternatives: 9

Speedup: 1.2×

• 20.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / 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 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / 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 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ -2 + \frac{4 \cdot \left(x - y\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ -2.0 (/ (* 4.0 (- x y)) z)))

C

double code(double x, double y, double z) {
	return -2.0 + ((4.0 * (x - y)) / 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 = (-2.0d0) + ((4.0d0 * (x - y)) / z)
end function

Java

public static double code(double x, double y, double z) {
	return -2.0 + ((4.0 * (x - y)) / z);
}

Python

def code(x, y, z):
	return -2.0 + ((4.0 * (x - y)) / z)

Julia

function code(x, y, z)
	return Float64(-2.0 + Float64(Float64(4.0 * Float64(x - y)) / z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = -2.0 + ((4.0 * (x - y)) / z);
end

Wolfram

code[x_, y_, z_] := N[(-2.0 + N[(N[(4.0 * N[(x - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

   2. sub-neg N/A

      \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

   4. *-commutative N/A

      \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

   9. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

   11. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

   14. *-inverses N/A

       \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

   17. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

   18. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

   21. --lowering--.f64 99.6%

       \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 52.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\frac{z}{4}}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-251}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+81}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ x (/ z 4.0))))
   (if (<= x -3.2e-12)
     t_0
     (if (<= x 1.3e-251) -2.0 (if (<= x 1.2e+81) (* -4.0 (/ y z)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x / (z / 4.0);
	double tmp;
	if (x <= -3.2e-12) {
		tmp = t_0;
	} else if (x <= 1.3e-251) {
		tmp = -2.0;
	} else if (x <= 1.2e+81) {
		tmp = -4.0 * (y / 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 = x / (z / 4.0d0)
    if (x <= (-3.2d-12)) then
        tmp = t_0
    else if (x <= 1.3d-251) then
        tmp = -2.0d0
    else if (x <= 1.2d+81) then
        tmp = (-4.0d0) * (y / 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 = x / (z / 4.0);
	double tmp;
	if (x <= -3.2e-12) {
		tmp = t_0;
	} else if (x <= 1.3e-251) {
		tmp = -2.0;
	} else if (x <= 1.2e+81) {
		tmp = -4.0 * (y / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x / (z / 4.0)
	tmp = 0
	if x <= -3.2e-12:
		tmp = t_0
	elif x <= 1.3e-251:
		tmp = -2.0
	elif x <= 1.2e+81:
		tmp = -4.0 * (y / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x / Float64(z / 4.0))
	tmp = 0.0
	if (x <= -3.2e-12)
		tmp = t_0;
	elseif (x <= 1.3e-251)
		tmp = -2.0;
	elseif (x <= 1.2e+81)
		tmp = Float64(-4.0 * Float64(y / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x / (z / 4.0);
	tmp = 0.0;
	if (x <= -3.2e-12)
		tmp = t_0;
	elseif (x <= 1.3e-251)
		tmp = -2.0;
	elseif (x <= 1.2e+81)
		tmp = -4.0 * (y / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x / N[(z / 4.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e-12], t$95$0, If[LessEqual[x, 1.3e-251], -2.0, If[LessEqual[x, 1.2e+81], N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000001e-12 or 1.19999999999999995e81 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{1 \cdot x}{z} \]

      2. associate-*l/ N/A

         \[\leadsto 4 \cdot \left(\frac{1}{z} \cdot \color{blue}{x}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot \color{blue}{x} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(4 \cdot \frac{1}{z}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(4 \cdot \frac{1}{z}\right)}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{4 \cdot 1}{\color{blue}{z}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{4}{z}\right)\right) \]

      8. /-lowering-/.f64 71.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(4, \color{blue}{z}\right)\right) \]

   7. Simplified 71.6%

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

      1. clear-num N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{z}{4}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{z}{4}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{z}{4}\right)}\right) \]

      4. /-lowering-/.f64 71.9%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 71.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{z}{4}}} \]

   if -3.2000000000000001e-12 < x < 1.3e-251

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   6. Step-by-step derivation

   7. Simplified 54.1%

      \[\leadsto \color{blue}{-2} \]

   if 1.3e-251 < x < 1.19999999999999995e81

   1. Initial program 98.6%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 98.7%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(\frac{y}{z}\right)}\right) \]

      2. /-lowering-/.f64 52.4%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   7. Simplified 52.4%

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

Alternative 3: 52.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4}{\frac{z}{x}}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.3 \cdot 10^{-254}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 4.1 \cdot 10^{+82}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ 4.0 (/ z x))))
   (if (<= x -3.2e-12)
     t_0
     (if (<= x 2.3e-254) -2.0 (if (<= x 4.1e+82) (* -4.0 (/ y z)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 / (z / x);
	double tmp;
	if (x <= -3.2e-12) {
		tmp = t_0;
	} else if (x <= 2.3e-254) {
		tmp = -2.0;
	} else if (x <= 4.1e+82) {
		tmp = -4.0 * (y / 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 = 4.0d0 / (z / x)
    if (x <= (-3.2d-12)) then
        tmp = t_0
    else if (x <= 2.3d-254) then
        tmp = -2.0d0
    else if (x <= 4.1d+82) then
        tmp = (-4.0d0) * (y / 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 = 4.0 / (z / x);
	double tmp;
	if (x <= -3.2e-12) {
		tmp = t_0;
	} else if (x <= 2.3e-254) {
		tmp = -2.0;
	} else if (x <= 4.1e+82) {
		tmp = -4.0 * (y / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 / (z / x)
	tmp = 0
	if x <= -3.2e-12:
		tmp = t_0
	elif x <= 2.3e-254:
		tmp = -2.0
	elif x <= 4.1e+82:
		tmp = -4.0 * (y / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 / Float64(z / x))
	tmp = 0.0
	if (x <= -3.2e-12)
		tmp = t_0;
	elseif (x <= 2.3e-254)
		tmp = -2.0;
	elseif (x <= 4.1e+82)
		tmp = Float64(-4.0 * Float64(y / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 / (z / x);
	tmp = 0.0;
	if (x <= -3.2e-12)
		tmp = t_0;
	elseif (x <= 2.3e-254)
		tmp = -2.0;
	elseif (x <= 4.1e+82)
		tmp = -4.0 * (y / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 / N[(z / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e-12], t$95$0, If[LessEqual[x, 2.3e-254], -2.0, If[LessEqual[x, 4.1e+82], N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000001e-12 or 4.09999999999999995e82 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{1 \cdot x}{z} \]

      2. associate-*l/ N/A

         \[\leadsto 4 \cdot \left(\frac{1}{z} \cdot \color{blue}{x}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot \color{blue}{x} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(4 \cdot \frac{1}{z}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(4 \cdot \frac{1}{z}\right)}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{4 \cdot 1}{\color{blue}{z}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{4}{z}\right)\right) \]

      8. /-lowering-/.f64 71.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(4, \color{blue}{z}\right)\right) \]

   7. Simplified 71.6%

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

      1. clear-num N/A

         \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{z}{4}}} \]

      2. un-div-inv N/A

         \[\leadsto \frac{x}{\color{blue}{\frac{z}{4}}} \]

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

         \[\leadsto \mathsf{/.f64}\left(x, \color{blue}{\left(\frac{z}{4}\right)}\right) \]

      4. /-lowering-/.f64 71.9%

         \[\leadsto \mathsf{/.f64}\left(x, \mathsf{/.f64}\left(z, \color{blue}{4}\right)\right) \]

   9. Applied egg-rr 71.9%

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

       1. associate-/r/ N/A

          \[\leadsto \frac{x}{z} \cdot \color{blue}{4} \]

       2. *-commutative N/A

          \[\leadsto 4 \cdot \color{blue}{\frac{x}{z}} \]

       3. clear-num N/A

          \[\leadsto 4 \cdot \frac{1}{\color{blue}{\frac{z}{x}}} \]

       4. un-div-inv N/A

          \[\leadsto \frac{4}{\color{blue}{\frac{z}{x}}} \]

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

          \[\leadsto \mathsf{/.f64}\left(4, \color{blue}{\left(\frac{z}{x}\right)}\right) \]

       6. /-lowering-/.f64 71.7%

          \[\leadsto \mathsf{/.f64}\left(4, \mathsf{/.f64}\left(z, \color{blue}{x}\right)\right) \]

   11. Applied egg-rr 71.7%

       \[\leadsto \color{blue}{\frac{4}{\frac{z}{x}}} \]

   if -3.2000000000000001e-12 < x < 2.2999999999999999e-254

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   6. Step-by-step derivation

   7. Simplified 54.1%

      \[\leadsto \color{blue}{-2} \]

   if 2.2999999999999999e-254 < x < 4.09999999999999995e82

   1. Initial program 98.6%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 98.7%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(\frac{y}{z}\right)}\right) \]

      2. /-lowering-/.f64 52.4%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   7. Simplified 52.4%

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

Alternative 4: 52.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \frac{4}{z}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-255}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+82}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* x (/ 4.0 z))))
   (if (<= x -3.2e-12)
     t_0
     (if (<= x 7.5e-255) -2.0 (if (<= x 1.45e+82) (* -4.0 (/ y z)) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = x * (4.0 / z);
	double tmp;
	if (x <= -3.2e-12) {
		tmp = t_0;
	} else if (x <= 7.5e-255) {
		tmp = -2.0;
	} else if (x <= 1.45e+82) {
		tmp = -4.0 * (y / 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 = x * (4.0d0 / z)
    if (x <= (-3.2d-12)) then
        tmp = t_0
    else if (x <= 7.5d-255) then
        tmp = -2.0d0
    else if (x <= 1.45d+82) then
        tmp = (-4.0d0) * (y / 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 = x * (4.0 / z);
	double tmp;
	if (x <= -3.2e-12) {
		tmp = t_0;
	} else if (x <= 7.5e-255) {
		tmp = -2.0;
	} else if (x <= 1.45e+82) {
		tmp = -4.0 * (y / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (4.0 / z)
	tmp = 0
	if x <= -3.2e-12:
		tmp = t_0
	elif x <= 7.5e-255:
		tmp = -2.0
	elif x <= 1.45e+82:
		tmp = -4.0 * (y / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(4.0 / z))
	tmp = 0.0
	if (x <= -3.2e-12)
		tmp = t_0;
	elseif (x <= 7.5e-255)
		tmp = -2.0;
	elseif (x <= 1.45e+82)
		tmp = Float64(-4.0 * Float64(y / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (4.0 / z);
	tmp = 0.0;
	if (x <= -3.2e-12)
		tmp = t_0;
	elseif (x <= 7.5e-255)
		tmp = -2.0;
	elseif (x <= 1.45e+82)
		tmp = -4.0 * (y / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(4.0 / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e-12], t$95$0, If[LessEqual[x, 7.5e-255], -2.0, If[LessEqual[x, 1.45e+82], N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2000000000000001e-12 or 1.4500000000000001e82 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf

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

      1. *-lft-identity N/A

         \[\leadsto 4 \cdot \frac{1 \cdot x}{z} \]

      2. associate-*l/ N/A

         \[\leadsto 4 \cdot \left(\frac{1}{z} \cdot \color{blue}{x}\right) \]

      3. associate-*l* N/A

         \[\leadsto \left(4 \cdot \frac{1}{z}\right) \cdot \color{blue}{x} \]

      4. *-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left(4 \cdot \frac{1}{z}\right)} \]

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

         \[\leadsto \mathsf{*.f64}\left(x, \color{blue}{\left(4 \cdot \frac{1}{z}\right)}\right) \]

      6. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{4 \cdot 1}{\color{blue}{z}}\right)\right) \]

      7. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(x, \left(\frac{4}{z}\right)\right) \]

      8. /-lowering-/.f64 71.6%

         \[\leadsto \mathsf{*.f64}\left(x, \mathsf{/.f64}\left(4, \color{blue}{z}\right)\right) \]

   7. Simplified 71.6%

      \[\leadsto \color{blue}{x \cdot \frac{4}{z}} \]

   if -3.2000000000000001e-12 < x < 7.50000000000000029e-255

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   6. Step-by-step derivation

   7. Simplified 54.1%

      \[\leadsto \color{blue}{-2} \]

   if 7.50000000000000029e-255 < x < 1.4500000000000001e82

   1. Initial program 98.6%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 98.7%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 98.7%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(\frac{y}{z}\right)}\right) \]

      2. /-lowering-/.f64 52.4%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   7. Simplified 52.4%

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

Alternative 5: 85.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-61}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-8}:\\ \;\;\;\;-2 + x \cdot \frac{4}{z}\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{y \cdot -4}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.3e-61)
   (* 4.0 (/ (- x y) z))
   (if (<= y 2.3e-8) (+ -2.0 (* x (/ 4.0 z))) (+ -2.0 (/ (* y -4.0) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.3e-61) {
		tmp = 4.0 * ((x - y) / z);
	} else if (y <= 2.3e-8) {
		tmp = -2.0 + (x * (4.0 / z));
	} else {
		tmp = -2.0 + ((y * -4.0) / z);
	}
	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 (y <= (-1.3d-61)) then
        tmp = 4.0d0 * ((x - y) / z)
    else if (y <= 2.3d-8) then
        tmp = (-2.0d0) + (x * (4.0d0 / z))
    else
        tmp = (-2.0d0) + ((y * (-4.0d0)) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.3e-61) {
		tmp = 4.0 * ((x - y) / z);
	} else if (y <= 2.3e-8) {
		tmp = -2.0 + (x * (4.0 / z));
	} else {
		tmp = -2.0 + ((y * -4.0) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.3e-61:
		tmp = 4.0 * ((x - y) / z)
	elif y <= 2.3e-8:
		tmp = -2.0 + (x * (4.0 / z))
	else:
		tmp = -2.0 + ((y * -4.0) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.3e-61)
		tmp = Float64(4.0 * Float64(Float64(x - y) / z));
	elseif (y <= 2.3e-8)
		tmp = Float64(-2.0 + Float64(x * Float64(4.0 / z)));
	else
		tmp = Float64(-2.0 + Float64(Float64(y * -4.0) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.3e-61)
		tmp = 4.0 * ((x - y) / z);
	elseif (y <= 2.3e-8)
		tmp = -2.0 + (x * (4.0 / z));
	else
		tmp = -2.0 + ((y * -4.0) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.3e-61], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.3e-8], N[(-2.0 + N[(x * N[(4.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-2.0 + N[(N[(y * -4.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.30000000000000005e-61

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(4, \color{blue}{\left(\frac{x - y}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\left(x - y\right), \color{blue}{z}\right)\right) \]

      3. --lowering--.f64 84.8%

         \[\leadsto \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), z\right)\right) \]

   7. Simplified 84.8%

      \[\leadsto \color{blue}{4 \cdot \frac{x - y}{z}} \]

   if -1.30000000000000005e-61 < y < 2.3000000000000001e-8

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x}{z} - 2} \]
   6. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto 4 \cdot \frac{x}{z} + \color{blue}{\left(\mathsf{neg}\left(2\right)\right)} \]

      2. metadata-eval N/A

         \[\leadsto 4 \cdot \frac{x}{z} + -2 \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{x}{z}\right), \color{blue}{-2}\right) \]

      4. *-lft-identity N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{1 \cdot x}{z}\right), -2\right) \]

      5. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{1}{z} \cdot x\right)\right), -2\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\left(4 \cdot \frac{1}{z}\right) \cdot x\right), -2\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\left(x \cdot \left(4 \cdot \frac{1}{z}\right)\right), -2\right) \]

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

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(4 \cdot \frac{1}{z}\right)\right), -2\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{4 \cdot 1}{z}\right)\right), -2\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \left(\frac{4}{z}\right)\right), -2\right) \]

      11. /-lowering-/.f64 95.4%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(x, \mathsf{/.f64}\left(4, z\right)\right), -2\right) \]

   7. Simplified 95.4%

      \[\leadsto \color{blue}{x \cdot \frac{4}{z} + -2} \]

   if 2.3000000000000001e-8 < y

   1. Initial program 98.6%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 98.6%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 98.6%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\color{blue}{\left(-4 \cdot y\right)}, z\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 83.1%

         \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, y\right), z\right)\right) \]

   7. Simplified 83.1%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-61}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-8}:\\ \;\;\;\;-2 + x \cdot \frac{4}{z}\\ \mathbf{else}:\\ \;\;\;\;-2 + \frac{y \cdot -4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x - y}{z}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-12}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+111}:\\ \;\;\;\;-2 + \frac{y \cdot -4}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ (- x y) z))))
   (if (<= x -3.2e-12)
     t_0
     (if (<= x 4.4e+111) (+ -2.0 (/ (* y -4.0) z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * ((x - y) / z);
	double tmp;
	if (x <= -3.2e-12) {
		tmp = t_0;
	} else if (x <= 4.4e+111) {
		tmp = -2.0 + ((y * -4.0) / 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 = 4.0d0 * ((x - y) / z)
    if (x <= (-3.2d-12)) then
        tmp = t_0
    else if (x <= 4.4d+111) then
        tmp = (-2.0d0) + ((y * (-4.0d0)) / 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 = 4.0 * ((x - y) / z);
	double tmp;
	if (x <= -3.2e-12) {
		tmp = t_0;
	} else if (x <= 4.4e+111) {
		tmp = -2.0 + ((y * -4.0) / z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * ((x - y) / z)
	tmp = 0
	if x <= -3.2e-12:
		tmp = t_0
	elif x <= 4.4e+111:
		tmp = -2.0 + ((y * -4.0) / z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(Float64(x - y) / z))
	tmp = 0.0
	if (x <= -3.2e-12)
		tmp = t_0;
	elseif (x <= 4.4e+111)
		tmp = Float64(-2.0 + Float64(Float64(y * -4.0) / z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * ((x - y) / z);
	tmp = 0.0;
	if (x <= -3.2e-12)
		tmp = t_0;
	elseif (x <= 4.4e+111)
		tmp = -2.0 + ((y * -4.0) / z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e-12], t$95$0, If[LessEqual[x, 4.4e+111], N[(-2.0 + N[(N[(y * -4.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000001e-12 or 4.39999999999999997e111 < x

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(4, \color{blue}{\left(\frac{x - y}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\left(x - y\right), \color{blue}{z}\right)\right) \]

      3. --lowering--.f64 90.0%

         \[\leadsto \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), z\right)\right) \]

   7. Simplified 90.0%

      \[\leadsto \color{blue}{4 \cdot \frac{x - y}{z}} \]

   if -3.2000000000000001e-12 < x < 4.39999999999999997e111

   1. Initial program 99.3%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 99.3%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 99.3%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0

      \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\color{blue}{\left(-4 \cdot y\right)}, z\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 88.0%

         \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(-4, y\right), z\right)\right) \]

   7. Simplified 88.0%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-12}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{+111}:\\ \;\;\;\;-2 + \frac{y \cdot -4}{z}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{+170}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+129}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.3e+170) -2.0 (if (<= z 9.5e+129) (* 4.0 (/ (- x y) z)) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.3e+170) {
		tmp = -2.0;
	} else if (z <= 9.5e+129) {
		tmp = 4.0 * ((x - y) / z);
	} else {
		tmp = -2.0;
	}
	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) :: tmp
    if (z <= (-1.3d+170)) then
        tmp = -2.0d0
    else if (z <= 9.5d+129) then
        tmp = 4.0d0 * ((x - y) / z)
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.3e+170) {
		tmp = -2.0;
	} else if (z <= 9.5e+129) {
		tmp = 4.0 * ((x - y) / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.3e+170:
		tmp = -2.0
	elif z <= 9.5e+129:
		tmp = 4.0 * ((x - y) / z)
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.3e+170)
		tmp = -2.0;
	elseif (z <= 9.5e+129)
		tmp = Float64(4.0 * Float64(Float64(x - y) / z));
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.3e+170)
		tmp = -2.0;
	elseif (z <= 9.5e+129)
		tmp = 4.0 * ((x - y) / z);
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.3e+170], -2.0, If[LessEqual[z, 9.5e+129], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2999999999999999e170 or 9.5000000000000004e129 < z

   1. Initial program 98.4%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 98.4%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 98.4%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   6. Step-by-step derivation

   7. Simplified 75.0%

      \[\leadsto \color{blue}{-2} \]

   if -1.2999999999999999e170 < z < 9.5000000000000004e129

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0

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

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

         \[\leadsto \mathsf{*.f64}\left(4, \color{blue}{\left(\frac{x - y}{z}\right)}\right) \]

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

         \[\leadsto \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\left(x - y\right), \color{blue}{z}\right)\right) \]

      3. --lowering--.f64 83.5%

         \[\leadsto \mathsf{*.f64}\left(4, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, y\right), z\right)\right) \]

   7. Simplified 83.5%

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

Alternative 8: 53.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+50}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-24}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -4.0 (/ y z))))
   (if (<= y -5.5e+50) t_0 (if (<= y 4.2e-24) -2.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = -4.0 * (y / z);
	double tmp;
	if (y <= -5.5e+50) {
		tmp = t_0;
	} else if (y <= 4.2e-24) {
		tmp = -2.0;
	} 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 = (-4.0d0) * (y / z)
    if (y <= (-5.5d+50)) then
        tmp = t_0
    else if (y <= 4.2d-24) then
        tmp = -2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -4.0 * (y / z);
	double tmp;
	if (y <= -5.5e+50) {
		tmp = t_0;
	} else if (y <= 4.2e-24) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -4.0 * (y / z)
	tmp = 0
	if y <= -5.5e+50:
		tmp = t_0
	elif y <= 4.2e-24:
		tmp = -2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-4.0 * Float64(y / z))
	tmp = 0.0
	if (y <= -5.5e+50)
		tmp = t_0;
	elseif (y <= 4.2e-24)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -4.0 * (y / z);
	tmp = 0.0;
	if (y <= -5.5e+50)
		tmp = t_0;
	elseif (y <= 4.2e-24)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5.5e+50], t$95$0, If[LessEqual[y, 4.2e-24], -2.0, t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4999999999999998e50 or 4.1999999999999999e-24 < y

   1. Initial program 99.2%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 99.2%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 99.2%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf

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

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

         \[\leadsto \mathsf{*.f64}\left(-4, \color{blue}{\left(\frac{y}{z}\right)}\right) \]

      2. /-lowering-/.f64 65.2%

         \[\leadsto \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(y, \color{blue}{z}\right)\right) \]

   7. Simplified 65.2%

      \[\leadsto \color{blue}{-4 \cdot \frac{y}{z}} \]

   if -5.4999999999999998e50 < y < 4.1999999999999999e-24

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Step-by-step derivation

      1. associate-*l/ N/A

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

      2. sub-neg N/A

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

      3. distribute-rgt-in N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

      4. *-commutative N/A

         \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

      5. +-commutative N/A

         \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

      7. associate-*l/ N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      8. associate-/l* N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      9. distribute-neg-frac N/A

         \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      11. associate-/l* N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      14. *-inverses N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

      15. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

      18. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

      19. *-commutative N/A

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

      21. --lowering--.f64 100.0%

          \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   6. Step-by-step derivation

   7. Simplified 43.5%

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

Alternative 9: 34.7% accurate, 11.0× speedup

Math

\[\begin{array}{l} \\ -2 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 -2.0)

C

double code(double x, double y, double z) {
	return -2.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return -2.0;
}

Python

def code(x, y, z):
	return -2.0

Julia

function code(x, y, z)
	return -2.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = -2.0;
end

Wolfram

code[x_, y_, z_] := -2.0

Derivation

1. Initial program 99.6%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) - z \cdot \frac{1}{2}\right)} \]

   2. sub-neg N/A

      \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}\right) \]

   3. distribute-rgt-in N/A

      \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) \cdot \frac{4}{z}} \]

   4. *-commutative N/A

      \[\leadsto \left(x - y\right) \cdot \frac{4}{z} + \frac{4}{z} \cdot \color{blue}{\left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right) + \color{blue}{\left(x - y\right) \cdot \frac{4}{z}} \]

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

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{4}{z} \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)\right), \color{blue}{\left(\left(x - y\right) \cdot \frac{4}{z}\right)}\right) \]

   7. associate-*l/ N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{4 \cdot \left(\mathsf{neg}\left(z \cdot \frac{1}{2}\right)\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

   8. associate-/l* N/A

      \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{\mathsf{neg}\left(z \cdot \frac{1}{2}\right)}{z}\right), \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

   9. distribute-neg-frac N/A

      \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{z \cdot \frac{1}{2}}{z}\right)\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2} \cdot z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

   11. associate-/l* N/A

       \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\mathsf{neg}\left(\frac{1}{2} \cdot \frac{z}{z}\right)\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{z}{z}\right)\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot \frac{z}{z}\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

   14. *-inverses N/A

       \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \left(\frac{-1}{2} \cdot 1\right)\right), \left(\left(x - y\right) \cdot \frac{4}{z}\right)\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(\left(4 \cdot \frac{-1}{2}\right), \left(\left(x - \color{blue}{y}\right) \cdot \frac{4}{z}\right)\right) \]

   16. metadata-eval N/A

       \[\leadsto \mathsf{+.f64}\left(-2, \left(\color{blue}{\left(x - y\right)} \cdot \frac{4}{z}\right)\right) \]

   17. associate-*r/ N/A

       \[\leadsto \mathsf{+.f64}\left(-2, \left(\frac{\left(x - y\right) \cdot 4}{\color{blue}{z}}\right)\right) \]

   18. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(\left(x - y\right) \cdot 4\right), \color{blue}{z}\right)\right) \]

   19. *-commutative N/A

       \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\left(4 \cdot \left(x - y\right)\right), z\right)\right) \]

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

       \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \left(x - y\right)\right), z\right)\right) \]

   21. --lowering--.f64 99.6%

       \[\leadsto \mathsf{+.f64}\left(-2, \mathsf{/.f64}\left(\mathsf{*.f64}\left(4, \mathsf{\_.f64}\left(x, y\right)\right), z\right)\right) \]

3. Simplified 99.6%

   \[\leadsto \color{blue}{-2 + \frac{4 \cdot \left(x - y\right)}{z}} \]
4. Add Preprocessing

5. Taylor expanded in z around inf

   \[\leadsto \color{blue}{-2} \]
6. Step-by-step derivation

7. Simplified 31.0%

   \[\leadsto \color{blue}{-2} \]
8. Add Preprocessing

Developer Target 1: 98.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{x}{z} - \left(2 + 4 \cdot \frac{y}{z}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* 4.0 (/ x z)) (+ 2.0 (* 4.0 (/ y z)))))

C

double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / 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 = (4.0d0 * (x / z)) - (2.0d0 + (4.0d0 * (y / z)))
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
}

Python

def code(x, y, z):
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)))

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(x / z)) - Float64(2.0 + Float64(4.0 * Float64(y / z))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] - N[(2.0 + N[(4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024138 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* 4 (/ x z)) (+ 2 (* 4 (/ y z)))))

  (/ (* 4.0 (- (- x y) (* z 0.5))) z))