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

Percentage Accurate: 99.8% → 100.0%

Time: 6.5s

Alternatives: 10

Speedup: 1.2×

• 21.1% 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: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{x - y}{z} - 2 \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - 2.0), $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/ 99.7%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

   2. sub-neg 99.7%

      \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

   3. distribute-rgt-neg-in 99.7%

      \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

   4. metadata-eval 99.7%

      \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

4. Taylor expanded in z around 0 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 53.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{-4}{z}\\ t_1 := x \cdot \frac{4}{z}\\ \mathbf{if}\;x \leq -7.6 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 10^{-215}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-116}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-44}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+105}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ -4.0 z))) (t_1 (* x (/ 4.0 z))))
   (if (<= x -7.6e-7)
     t_1
     (if (<= x 1e-215)
       -2.0
       (if (<= x 1.05e-116)
         t_0
         (if (<= x 2.5e-44) -2.0 (if (<= x 3.1e+105) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (-4.0 / z);
	double t_1 = x * (4.0 / z);
	double tmp;
	if (x <= -7.6e-7) {
		tmp = t_1;
	} else if (x <= 1e-215) {
		tmp = -2.0;
	} else if (x <= 1.05e-116) {
		tmp = t_0;
	} else if (x <= 2.5e-44) {
		tmp = -2.0;
	} else if (x <= 3.1e+105) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = y * ((-4.0d0) / z)
    t_1 = x * (4.0d0 / z)
    if (x <= (-7.6d-7)) then
        tmp = t_1
    else if (x <= 1d-215) then
        tmp = -2.0d0
    else if (x <= 1.05d-116) then
        tmp = t_0
    else if (x <= 2.5d-44) then
        tmp = -2.0d0
    else if (x <= 3.1d+105) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (-4.0 / z);
	double t_1 = x * (4.0 / z);
	double tmp;
	if (x <= -7.6e-7) {
		tmp = t_1;
	} else if (x <= 1e-215) {
		tmp = -2.0;
	} else if (x <= 1.05e-116) {
		tmp = t_0;
	} else if (x <= 2.5e-44) {
		tmp = -2.0;
	} else if (x <= 3.1e+105) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (-4.0 / z)
	t_1 = x * (4.0 / z)
	tmp = 0
	if x <= -7.6e-7:
		tmp = t_1
	elif x <= 1e-215:
		tmp = -2.0
	elif x <= 1.05e-116:
		tmp = t_0
	elif x <= 2.5e-44:
		tmp = -2.0
	elif x <= 3.1e+105:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(-4.0 / z))
	t_1 = Float64(x * Float64(4.0 / z))
	tmp = 0.0
	if (x <= -7.6e-7)
		tmp = t_1;
	elseif (x <= 1e-215)
		tmp = -2.0;
	elseif (x <= 1.05e-116)
		tmp = t_0;
	elseif (x <= 2.5e-44)
		tmp = -2.0;
	elseif (x <= 3.1e+105)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (-4.0 / z);
	t_1 = x * (4.0 / z);
	tmp = 0.0;
	if (x <= -7.6e-7)
		tmp = t_1;
	elseif (x <= 1e-215)
		tmp = -2.0;
	elseif (x <= 1.05e-116)
		tmp = t_0;
	elseif (x <= 2.5e-44)
		tmp = -2.0;
	elseif (x <= 3.1e+105)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(-4.0 / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(4.0 / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.6e-7], t$95$1, If[LessEqual[x, 1e-215], -2.0, If[LessEqual[x, 1.05e-116], t$95$0, If[LessEqual[x, 2.5e-44], -2.0, If[LessEqual[x, 3.1e+105], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.60000000000000029e-7 or 3.10000000000000004e105 < x

   1. Initial program 99.0%

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

      1. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.6%

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

      1. associate-+l- 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(x - \left(y - z \cdot -0.5\right)\right)} \]

      2. *-commutative 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y - \color{blue}{-0.5 \cdot z}\right)\right) \]

      3. cancel-sign-sub-inv 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\left(y + \left(--0.5\right) \cdot z\right)}\right) \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y + \color{blue}{0.5} \cdot z\right)\right) \]

      5. *-commutative 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y + \color{blue}{z \cdot 0.5}\right)\right) \]

      6. +-commutative 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\left(z \cdot 0.5 + y\right)}\right) \]

      7. fma-udef 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\mathsf{fma}\left(z, 0.5, y\right)}\right) \]

      8. associate-/r/ 99.7%

         \[\leadsto \color{blue}{\frac{4}{\frac{z}{x - \mathsf{fma}\left(z, 0.5, y\right)}}} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{4}{\frac{z}{x - \mathsf{fma}\left(z, 0.5, y\right)}}} \]

   6. Taylor expanded in x around inf 76.0%

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

      1. associate-/r/ 76.0%

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

   8. Applied egg-rr 76.0%

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

   if -7.60000000000000029e-7 < x < 1.00000000000000004e-215 or 1.05e-116 < x < 2.50000000000000019e-44

   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/ 99.8%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.8%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in z around inf 55.3%

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

   if 1.00000000000000004e-215 < x < 1.05e-116 or 2.50000000000000019e-44 < x < 3.10000000000000004e105

   1. Initial program 99.9%

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

      1. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.8%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.8%

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

      1. associate-+l- 99.8%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(x - \left(y - z \cdot -0.5\right)\right)} \]

      2. *-commutative 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y - \color{blue}{-0.5 \cdot z}\right)\right) \]

      3. cancel-sign-sub-inv 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\left(y + \left(--0.5\right) \cdot z\right)}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y + \color{blue}{0.5} \cdot z\right)\right) \]

      5. *-commutative 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y + \color{blue}{z \cdot 0.5}\right)\right) \]

      6. +-commutative 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\left(z \cdot 0.5 + y\right)}\right) \]

      7. fma-udef 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\mathsf{fma}\left(z, 0.5, y\right)}\right) \]

      8. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{4}{\frac{z}{x - \mathsf{fma}\left(z, 0.5, y\right)}}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{4}{\frac{z}{x - \mathsf{fma}\left(z, 0.5, y\right)}}} \]

   6. Taylor expanded in z around 0 99.9%

      \[\leadsto \frac{4}{\frac{z}{\color{blue}{\left(x + -0.5 \cdot z\right) - y}}} \]

   7. Taylor expanded in y around inf 58.8%

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

      1. associate-*r/ 58.8%

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

      2. *-rgt-identity 58.8%

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

      3. times-frac 58.7%

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

      4. /-rgt-identity 58.7%

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

   9. Simplified 58.7%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-7}:\\ \;\;\;\;x \cdot \frac{4}{z}\\ \mathbf{elif}\;x \leq 10^{-215}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 1.05 \cdot 10^{-116}:\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-44}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{4}{z}\\ \end{array} \]

Alternative 3: 53.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := y \cdot \frac{-4}{z}\\ t_1 := 4 \cdot \frac{x}{z}\\ \mathbf{if}\;x \leq -4.7 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-215}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-113}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-47}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+105}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* y (/ -4.0 z))) (t_1 (* 4.0 (/ x z))))
   (if (<= x -4.7e-6)
     t_1
     (if (<= x 1.5e-215)
       -2.0
       (if (<= x 4.4e-113)
         t_0
         (if (<= x 3.5e-47) -2.0 (if (<= x 3.8e+105) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = y * (-4.0 / z);
	double t_1 = 4.0 * (x / z);
	double tmp;
	if (x <= -4.7e-6) {
		tmp = t_1;
	} else if (x <= 1.5e-215) {
		tmp = -2.0;
	} else if (x <= 4.4e-113) {
		tmp = t_0;
	} else if (x <= 3.5e-47) {
		tmp = -2.0;
	} else if (x <= 3.8e+105) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = y * ((-4.0d0) / z)
    t_1 = 4.0d0 * (x / z)
    if (x <= (-4.7d-6)) then
        tmp = t_1
    else if (x <= 1.5d-215) then
        tmp = -2.0d0
    else if (x <= 4.4d-113) then
        tmp = t_0
    else if (x <= 3.5d-47) then
        tmp = -2.0d0
    else if (x <= 3.8d+105) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = y * (-4.0 / z);
	double t_1 = 4.0 * (x / z);
	double tmp;
	if (x <= -4.7e-6) {
		tmp = t_1;
	} else if (x <= 1.5e-215) {
		tmp = -2.0;
	} else if (x <= 4.4e-113) {
		tmp = t_0;
	} else if (x <= 3.5e-47) {
		tmp = -2.0;
	} else if (x <= 3.8e+105) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = y * (-4.0 / z)
	t_1 = 4.0 * (x / z)
	tmp = 0
	if x <= -4.7e-6:
		tmp = t_1
	elif x <= 1.5e-215:
		tmp = -2.0
	elif x <= 4.4e-113:
		tmp = t_0
	elif x <= 3.5e-47:
		tmp = -2.0
	elif x <= 3.8e+105:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(y * Float64(-4.0 / z))
	t_1 = Float64(4.0 * Float64(x / z))
	tmp = 0.0
	if (x <= -4.7e-6)
		tmp = t_1;
	elseif (x <= 1.5e-215)
		tmp = -2.0;
	elseif (x <= 4.4e-113)
		tmp = t_0;
	elseif (x <= 3.5e-47)
		tmp = -2.0;
	elseif (x <= 3.8e+105)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = y * (-4.0 / z);
	t_1 = 4.0 * (x / z);
	tmp = 0.0;
	if (x <= -4.7e-6)
		tmp = t_1;
	elseif (x <= 1.5e-215)
		tmp = -2.0;
	elseif (x <= 4.4e-113)
		tmp = t_0;
	elseif (x <= 3.5e-47)
		tmp = -2.0;
	elseif (x <= 3.8e+105)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(y * N[(-4.0 / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.7e-6], t$95$1, If[LessEqual[x, 1.5e-215], -2.0, If[LessEqual[x, 4.4e-113], t$95$0, If[LessEqual[x, 3.5e-47], -2.0, If[LessEqual[x, 3.8e+105], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.69999999999999989e-6 or 3.8e105 < x

   1. Initial program 99.0%

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

      1. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in x around inf 76.2%

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

      1. *-commutative 76.2%

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

   6. Simplified 76.2%

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

   if -4.69999999999999989e-6 < x < 1.50000000000000013e-215 or 4.40000000000000008e-113 < x < 3.4999999999999998e-47

   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/ 99.8%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.8%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in z around inf 55.3%

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

   if 1.50000000000000013e-215 < x < 4.40000000000000008e-113 or 3.4999999999999998e-47 < x < 3.8e105

   1. Initial program 99.9%

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

      1. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.8%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.8%

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

      1. associate-+l- 99.8%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(x - \left(y - z \cdot -0.5\right)\right)} \]

      2. *-commutative 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y - \color{blue}{-0.5 \cdot z}\right)\right) \]

      3. cancel-sign-sub-inv 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\left(y + \left(--0.5\right) \cdot z\right)}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y + \color{blue}{0.5} \cdot z\right)\right) \]

      5. *-commutative 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y + \color{blue}{z \cdot 0.5}\right)\right) \]

      6. +-commutative 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\left(z \cdot 0.5 + y\right)}\right) \]

      7. fma-udef 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\mathsf{fma}\left(z, 0.5, y\right)}\right) \]

      8. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{4}{\frac{z}{x - \mathsf{fma}\left(z, 0.5, y\right)}}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{4}{\frac{z}{x - \mathsf{fma}\left(z, 0.5, y\right)}}} \]

   6. Taylor expanded in z around 0 99.9%

      \[\leadsto \frac{4}{\frac{z}{\color{blue}{\left(x + -0.5 \cdot z\right) - y}}} \]

   7. Taylor expanded in y around inf 58.8%

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

      1. associate-*r/ 58.8%

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

      2. *-rgt-identity 58.8%

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

      3. times-frac 58.7%

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

      4. /-rgt-identity 58.7%

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

   9. Simplified 58.7%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.7 \cdot 10^{-6}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-215}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-113}:\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-47}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+105}:\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \end{array} \]

Alternative 4: 53.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{z}\\ \mathbf{if}\;x \leq -7.8 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-208}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 10^{-111}:\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-43}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+106}:\\ \;\;\;\;\frac{-4}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x z))))
   (if (<= x -7.8e-7)
     t_0
     (if (<= x 5.2e-208)
       -2.0
       (if (<= x 1e-111)
         (* y (/ -4.0 z))
         (if (<= x 2.1e-43) -2.0 (if (<= x 3e+106) (/ -4.0 (/ z y)) t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / z);
	double tmp;
	if (x <= -7.8e-7) {
		tmp = t_0;
	} else if (x <= 5.2e-208) {
		tmp = -2.0;
	} else if (x <= 1e-111) {
		tmp = y * (-4.0 / z);
	} else if (x <= 2.1e-43) {
		tmp = -2.0;
	} else if (x <= 3e+106) {
		tmp = -4.0 / (z / y);
	} 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 / z)
    if (x <= (-7.8d-7)) then
        tmp = t_0
    else if (x <= 5.2d-208) then
        tmp = -2.0d0
    else if (x <= 1d-111) then
        tmp = y * ((-4.0d0) / z)
    else if (x <= 2.1d-43) then
        tmp = -2.0d0
    else if (x <= 3d+106) then
        tmp = (-4.0d0) / (z / y)
    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 / z);
	double tmp;
	if (x <= -7.8e-7) {
		tmp = t_0;
	} else if (x <= 5.2e-208) {
		tmp = -2.0;
	} else if (x <= 1e-111) {
		tmp = y * (-4.0 / z);
	} else if (x <= 2.1e-43) {
		tmp = -2.0;
	} else if (x <= 3e+106) {
		tmp = -4.0 / (z / y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / z)
	tmp = 0
	if x <= -7.8e-7:
		tmp = t_0
	elif x <= 5.2e-208:
		tmp = -2.0
	elif x <= 1e-111:
		tmp = y * (-4.0 / z)
	elif x <= 2.1e-43:
		tmp = -2.0
	elif x <= 3e+106:
		tmp = -4.0 / (z / y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / z))
	tmp = 0.0
	if (x <= -7.8e-7)
		tmp = t_0;
	elseif (x <= 5.2e-208)
		tmp = -2.0;
	elseif (x <= 1e-111)
		tmp = Float64(y * Float64(-4.0 / z));
	elseif (x <= 2.1e-43)
		tmp = -2.0;
	elseif (x <= 3e+106)
		tmp = Float64(-4.0 / Float64(z / y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / z);
	tmp = 0.0;
	if (x <= -7.8e-7)
		tmp = t_0;
	elseif (x <= 5.2e-208)
		tmp = -2.0;
	elseif (x <= 1e-111)
		tmp = y * (-4.0 / z);
	elseif (x <= 2.1e-43)
		tmp = -2.0;
	elseif (x <= 3e+106)
		tmp = -4.0 / (z / y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.8e-7], t$95$0, If[LessEqual[x, 5.2e-208], -2.0, If[LessEqual[x, 1e-111], N[(y * N[(-4.0 / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.1e-43], -2.0, If[LessEqual[x, 3e+106], N[(-4.0 / N[(z / y), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -7.80000000000000049e-7 or 3.0000000000000001e106 < x

   1. Initial program 99.0%

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

      1. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in x around inf 76.2%

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

      1. *-commutative 76.2%

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

   6. Simplified 76.2%

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

   if -7.80000000000000049e-7 < x < 5.20000000000000034e-208 or 1.00000000000000009e-111 < x < 2.1000000000000001e-43

   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/ 99.8%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.8%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in z around inf 55.3%

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

   if 5.20000000000000034e-208 < x < 1.00000000000000009e-111

   1. Initial program 99.9%

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

      1. associate-*l/ 99.7%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.7%

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

      1. associate-+l- 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(x - \left(y - z \cdot -0.5\right)\right)} \]

      2. *-commutative 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y - \color{blue}{-0.5 \cdot z}\right)\right) \]

      3. cancel-sign-sub-inv 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\left(y + \left(--0.5\right) \cdot z\right)}\right) \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y + \color{blue}{0.5} \cdot z\right)\right) \]

      5. *-commutative 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y + \color{blue}{z \cdot 0.5}\right)\right) \]

      6. +-commutative 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\left(z \cdot 0.5 + y\right)}\right) \]

      7. fma-udef 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\mathsf{fma}\left(z, 0.5, y\right)}\right) \]

      8. associate-/r/ 99.8%

         \[\leadsto \color{blue}{\frac{4}{\frac{z}{x - \mathsf{fma}\left(z, 0.5, y\right)}}} \]

   5. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{4}{\frac{z}{x - \mathsf{fma}\left(z, 0.5, y\right)}}} \]

   6. Taylor expanded in z around 0 99.8%

      \[\leadsto \frac{4}{\frac{z}{\color{blue}{\left(x + -0.5 \cdot z\right) - y}}} \]

   7. Taylor expanded in y around inf 63.7%

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

      1. associate-*r/ 63.7%

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

      2. *-rgt-identity 63.7%

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

      3. times-frac 63.7%

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

      4. /-rgt-identity 63.7%

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

   9. Simplified 63.7%

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

   if 2.1000000000000001e-43 < x < 3.0000000000000001e106

   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/ 99.8%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.8%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.8%

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

      1. associate-+l- 99.8%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(x - \left(y - z \cdot -0.5\right)\right)} \]

      2. *-commutative 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y - \color{blue}{-0.5 \cdot z}\right)\right) \]

      3. cancel-sign-sub-inv 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\left(y + \left(--0.5\right) \cdot z\right)}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y + \color{blue}{0.5} \cdot z\right)\right) \]

      5. *-commutative 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y + \color{blue}{z \cdot 0.5}\right)\right) \]

      6. +-commutative 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\left(z \cdot 0.5 + y\right)}\right) \]

      7. fma-udef 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\mathsf{fma}\left(z, 0.5, y\right)}\right) \]

      8. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{4}{\frac{z}{x - \mathsf{fma}\left(z, 0.5, y\right)}}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{4}{\frac{z}{x - \mathsf{fma}\left(z, 0.5, y\right)}}} \]

   6. Taylor expanded in z around 0 99.9%

      \[\leadsto \frac{4}{\frac{z}{\color{blue}{\left(x + -0.5 \cdot z\right) - y}}} \]

   7. Taylor expanded in y around inf 55.2%

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

      1. associate-*r/ 55.2%

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

      2. *-rgt-identity 55.2%

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

      3. times-frac 55.0%

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

      4. /-rgt-identity 55.0%

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

   9. Simplified 55.0%

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

       1. associate-*l/ 55.2%

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

       2. associate-/l* 55.2%

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

   11. Applied egg-rr 55.2%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.8 \cdot 10^{-7}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{elif}\;x \leq 5.2 \cdot 10^{-208}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 10^{-111}:\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-43}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+106}:\\ \;\;\;\;\frac{-4}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \end{array} \]

Alternative 5: 53.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y \cdot -4}{z}\\ t_1 := 4 \cdot \frac{x}{z}\\ \mathbf{if}\;x \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-211}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-43}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* y -4.0) z)) (t_1 (* 4.0 (/ x z))))
   (if (<= x -3.4e-8)
     t_1
     (if (<= x 1.2e-211)
       -2.0
       (if (<= x 9.8e-114)
         t_0
         (if (<= x 3.3e-43) -2.0 (if (<= x 3e+106) t_0 t_1)))))))

C

double code(double x, double y, double z) {
	double t_0 = (y * -4.0) / z;
	double t_1 = 4.0 * (x / z);
	double tmp;
	if (x <= -3.4e-8) {
		tmp = t_1;
	} else if (x <= 1.2e-211) {
		tmp = -2.0;
	} else if (x <= 9.8e-114) {
		tmp = t_0;
	} else if (x <= 3.3e-43) {
		tmp = -2.0;
	} else if (x <= 3e+106) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = (y * (-4.0d0)) / z
    t_1 = 4.0d0 * (x / z)
    if (x <= (-3.4d-8)) then
        tmp = t_1
    else if (x <= 1.2d-211) then
        tmp = -2.0d0
    else if (x <= 9.8d-114) then
        tmp = t_0
    else if (x <= 3.3d-43) then
        tmp = -2.0d0
    else if (x <= 3d+106) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y * -4.0) / z;
	double t_1 = 4.0 * (x / z);
	double tmp;
	if (x <= -3.4e-8) {
		tmp = t_1;
	} else if (x <= 1.2e-211) {
		tmp = -2.0;
	} else if (x <= 9.8e-114) {
		tmp = t_0;
	} else if (x <= 3.3e-43) {
		tmp = -2.0;
	} else if (x <= 3e+106) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y * -4.0) / z
	t_1 = 4.0 * (x / z)
	tmp = 0
	if x <= -3.4e-8:
		tmp = t_1
	elif x <= 1.2e-211:
		tmp = -2.0
	elif x <= 9.8e-114:
		tmp = t_0
	elif x <= 3.3e-43:
		tmp = -2.0
	elif x <= 3e+106:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y * -4.0) / z)
	t_1 = Float64(4.0 * Float64(x / z))
	tmp = 0.0
	if (x <= -3.4e-8)
		tmp = t_1;
	elseif (x <= 1.2e-211)
		tmp = -2.0;
	elseif (x <= 9.8e-114)
		tmp = t_0;
	elseif (x <= 3.3e-43)
		tmp = -2.0;
	elseif (x <= 3e+106)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y * -4.0) / z;
	t_1 = 4.0 * (x / z);
	tmp = 0.0;
	if (x <= -3.4e-8)
		tmp = t_1;
	elseif (x <= 1.2e-211)
		tmp = -2.0;
	elseif (x <= 9.8e-114)
		tmp = t_0;
	elseif (x <= 3.3e-43)
		tmp = -2.0;
	elseif (x <= 3e+106)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y * -4.0), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.4e-8], t$95$1, If[LessEqual[x, 1.2e-211], -2.0, If[LessEqual[x, 9.8e-114], t$95$0, If[LessEqual[x, 3.3e-43], -2.0, If[LessEqual[x, 3e+106], t$95$0, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4e-8 or 3.0000000000000001e106 < x

   1. Initial program 99.0%

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

      1. associate-*l/ 99.6%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in x around inf 76.2%

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

      1. *-commutative 76.2%

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

   6. Simplified 76.2%

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

   if -3.4e-8 < x < 1.2000000000000001e-211 or 9.7999999999999994e-114 < x < 3.30000000000000016e-43

   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/ 99.8%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.8%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in z around inf 55.3%

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

   if 1.2000000000000001e-211 < x < 9.7999999999999994e-114 or 3.30000000000000016e-43 < x < 3.0000000000000001e106

   1. Initial program 99.9%

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

      1. associate-*l/ 99.8%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.8%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in y around inf 58.8%

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

      1. associate-*r/ 58.8%

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

   6. Simplified 58.8%

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

4. Final simplification 64.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-8}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-211}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 9.8 \cdot 10^{-114}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-43}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+106}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \end{array} \]

Alternative 6: 85.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-8} \lor \neg \left(x \leq 0.029\right):\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.6e-8) (not (<= x 0.029)))
   (* 4.0 (/ (- x y) z))
   (- (* -4.0 (/ y z)) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.6e-8) || !(x <= 0.029)) {
		tmp = 4.0 * ((x - y) / z);
	} else {
		tmp = (-4.0 * (y / z)) - 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 ((x <= (-8.6d-8)) .or. (.not. (x <= 0.029d0))) then
        tmp = 4.0d0 * ((x - y) / z)
    else
        tmp = ((-4.0d0) * (y / z)) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.6e-8) || !(x <= 0.029)) {
		tmp = 4.0 * ((x - y) / z);
	} else {
		tmp = (-4.0 * (y / z)) - 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.6e-8) or not (x <= 0.029):
		tmp = 4.0 * ((x - y) / z)
	else:
		tmp = (-4.0 * (y / z)) - 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.6e-8) || !(x <= 0.029))
		tmp = Float64(4.0 * Float64(Float64(x - y) / z));
	else
		tmp = Float64(Float64(-4.0 * Float64(y / z)) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.6e-8) || ~((x <= 0.029)))
		tmp = 4.0 * ((x - y) / z);
	else
		tmp = (-4.0 * (y / z)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.6e-8], N[Not[LessEqual[x, 0.029]], $MachinePrecision]], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.6000000000000002e-8 or 0.0290000000000000015 < x

   1. Initial program 99.1%

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

      1. associate-*r/ 100.0%

         \[\leadsto \color{blue}{4 \cdot \frac{\left(x - y\right) - z \cdot 0.5}{z}} \]

      2. associate--l- 100.0%

         \[\leadsto 4 \cdot \frac{\color{blue}{x - \left(y + z \cdot 0.5\right)}}{z} \]

      3. +-commutative 100.0%

         \[\leadsto 4 \cdot \frac{x - \color{blue}{\left(z \cdot 0.5 + y\right)}}{z} \]

      4. fma-def 100.0%

         \[\leadsto 4 \cdot \frac{x - \color{blue}{\mathsf{fma}\left(z, 0.5, y\right)}}{z} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{4 \cdot \frac{x - \mathsf{fma}\left(z, 0.5, y\right)}{z}} \]

   4. Taylor expanded in z around 0 86.3%

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

   if -8.6000000000000002e-8 < x < 0.0290000000000000015

   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/ 99.7%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in z around 0 100.0%

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

   5. Taylor expanded in x around 0 89.5%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.6 \cdot 10^{-8} \lor \neg \left(x \leq 0.029\right):\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \end{array} \]

Alternative 7: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+59} \lor \neg \left(z \leq 4.5 \cdot 10^{-52}\right):\\ \;\;\;\;4 \cdot \frac{x}{z} - 2\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -2e+59) (not (<= z 4.5e-52)))
   (- (* 4.0 (/ x z)) 2.0)
   (* 4.0 (/ (- x y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2e+59) || !(z <= 4.5e-52)) {
		tmp = (4.0 * (x / z)) - 2.0;
	} else {
		tmp = 4.0 * ((x - y) / 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 ((z <= (-2d+59)) .or. (.not. (z <= 4.5d-52))) then
        tmp = (4.0d0 * (x / z)) - 2.0d0
    else
        tmp = 4.0d0 * ((x - y) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -2e+59) || !(z <= 4.5e-52)) {
		tmp = (4.0 * (x / z)) - 2.0;
	} else {
		tmp = 4.0 * ((x - y) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -2e+59) or not (z <= 4.5e-52):
		tmp = (4.0 * (x / z)) - 2.0
	else:
		tmp = 4.0 * ((x - y) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -2e+59) || !(z <= 4.5e-52))
		tmp = Float64(Float64(4.0 * Float64(x / z)) - 2.0);
	else
		tmp = Float64(4.0 * Float64(Float64(x - y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -2e+59) || ~((z <= 4.5e-52)))
		tmp = (4.0 * (x / z)) - 2.0;
	else
		tmp = 4.0 * ((x - y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -2e+59], N[Not[LessEqual[z, 4.5e-52]], $MachinePrecision]], N[(N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.99999999999999994e59 or 4.5e-52 < z

   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/ 99.6%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in z around 0 100.0%

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

   5. Taylor expanded in x around inf 82.8%

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

   if -1.99999999999999994e59 < z < 4.5e-52

   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-*r/ 100.0%

         \[\leadsto \color{blue}{4 \cdot \frac{\left(x - y\right) - z \cdot 0.5}{z}} \]

      2. associate--l- 100.0%

         \[\leadsto 4 \cdot \frac{\color{blue}{x - \left(y + z \cdot 0.5\right)}}{z} \]

      3. +-commutative 100.0%

         \[\leadsto 4 \cdot \frac{x - \color{blue}{\left(z \cdot 0.5 + y\right)}}{z} \]

      4. fma-def 100.0%

         \[\leadsto 4 \cdot \frac{x - \color{blue}{\mathsf{fma}\left(z, 0.5, y\right)}}{z} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{4 \cdot \frac{x - \mathsf{fma}\left(z, 0.5, y\right)}{z}} \]

   4. Taylor expanded in z around 0 94.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+59} \lor \neg \left(z \leq 4.5 \cdot 10^{-52}\right):\\ \;\;\;\;4 \cdot \frac{x}{z} - 2\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \end{array} \]

Alternative 8: 79.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+181}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+114}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.7e+181) -2.0 (if (<= z 1.4e+114) (* 4.0 (/ (- x y) z)) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.7e+181) {
		tmp = -2.0;
	} else if (z <= 1.4e+114) {
		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 <= (-2.7d+181)) then
        tmp = -2.0d0
    else if (z <= 1.4d+114) 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 <= -2.7e+181) {
		tmp = -2.0;
	} else if (z <= 1.4e+114) {
		tmp = 4.0 * ((x - y) / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.7e+181:
		tmp = -2.0
	elif z <= 1.4e+114:
		tmp = 4.0 * ((x - y) / z)
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.7e+181)
		tmp = -2.0;
	elseif (z <= 1.4e+114)
		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 <= -2.7e+181)
		tmp = -2.0;
	elseif (z <= 1.4e+114)
		tmp = 4.0 * ((x - y) / z);
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.7e+181], -2.0, If[LessEqual[z, 1.4e+114], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if z < -2.70000000000000007e181 or 1.4e114 < z

   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/ 99.6%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in z around inf 78.0%

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

   if -2.70000000000000007e181 < z < 1.4e114

   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-*r/ 100.0%

         \[\leadsto \color{blue}{4 \cdot \frac{\left(x - y\right) - z \cdot 0.5}{z}} \]

      2. associate--l- 100.0%

         \[\leadsto 4 \cdot \frac{\color{blue}{x - \left(y + z \cdot 0.5\right)}}{z} \]

      3. +-commutative 100.0%

         \[\leadsto 4 \cdot \frac{x - \color{blue}{\left(z \cdot 0.5 + y\right)}}{z} \]

      4. fma-def 100.0%

         \[\leadsto 4 \cdot \frac{x - \color{blue}{\mathsf{fma}\left(z, 0.5, y\right)}}{z} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{4 \cdot \frac{x - \mathsf{fma}\left(z, 0.5, y\right)}{z}} \]

   4. Taylor expanded in z around 0 85.2%

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

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.7 \cdot 10^{+181}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+114}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 9: 52.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+55}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-53}:\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2e+55) -2.0 (if (<= z 7.4e-53) (* y (/ -4.0 z)) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2e+55) {
		tmp = -2.0;
	} else if (z <= 7.4e-53) {
		tmp = y * (-4.0 / 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 <= (-2d+55)) then
        tmp = -2.0d0
    else if (z <= 7.4d-53) then
        tmp = y * ((-4.0d0) / 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 <= -2e+55) {
		tmp = -2.0;
	} else if (z <= 7.4e-53) {
		tmp = y * (-4.0 / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2e+55:
		tmp = -2.0
	elif z <= 7.4e-53:
		tmp = y * (-4.0 / z)
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2e+55)
		tmp = -2.0;
	elseif (z <= 7.4e-53)
		tmp = Float64(y * Float64(-4.0 / z));
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2e+55)
		tmp = -2.0;
	elseif (z <= 7.4e-53)
		tmp = y * (-4.0 / z);
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2e+55], -2.0, If[LessEqual[z, 7.4e-53], N[(y * N[(-4.0 / z), $MachinePrecision]), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if z < -2.00000000000000002e55 or 7.39999999999999965e-53 < z

   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/ 99.6%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.6%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.6%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.6%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

   4. Taylor expanded in z around inf 63.2%

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

   if -2.00000000000000002e55 < z < 7.39999999999999965e-53

   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/ 99.7%

         \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

      2. sub-neg 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

      3. distribute-rgt-neg-in 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

   3. Simplified 99.7%

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

      1. associate-+l- 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(x - \left(y - z \cdot -0.5\right)\right)} \]

      2. *-commutative 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y - \color{blue}{-0.5 \cdot z}\right)\right) \]

      3. cancel-sign-sub-inv 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\left(y + \left(--0.5\right) \cdot z\right)}\right) \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y + \color{blue}{0.5} \cdot z\right)\right) \]

      5. *-commutative 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(x - \left(y + \color{blue}{z \cdot 0.5}\right)\right) \]

      6. +-commutative 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\left(z \cdot 0.5 + y\right)}\right) \]

      7. fma-udef 99.7%

         \[\leadsto \frac{4}{z} \cdot \left(x - \color{blue}{\mathsf{fma}\left(z, 0.5, y\right)}\right) \]

      8. associate-/r/ 99.7%

         \[\leadsto \color{blue}{\frac{4}{\frac{z}{x - \mathsf{fma}\left(z, 0.5, y\right)}}} \]

   5. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{4}{\frac{z}{x - \mathsf{fma}\left(z, 0.5, y\right)}}} \]

   6. Taylor expanded in z around 0 99.7%

      \[\leadsto \frac{4}{\frac{z}{\color{blue}{\left(x + -0.5 \cdot z\right) - y}}} \]

   7. Taylor expanded in y around inf 47.9%

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

      1. associate-*r/ 47.9%

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

      2. *-rgt-identity 47.9%

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

      3. times-frac 47.8%

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

      4. /-rgt-identity 47.8%

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

   9. Simplified 47.8%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+55}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{-53}:\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 10: 34.0% 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/ 99.7%

      \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) - z \cdot 0.5\right)} \]

   2. sub-neg 99.7%

      \[\leadsto \frac{4}{z} \cdot \color{blue}{\left(\left(x - y\right) + \left(-z \cdot 0.5\right)\right)} \]

   3. distribute-rgt-neg-in 99.7%

      \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + \color{blue}{z \cdot \left(-0.5\right)}\right) \]

   4. metadata-eval 99.7%

      \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot \color{blue}{-0.5}\right) \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{\frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right)} \]

4. Taylor expanded in z around inf 33.0%

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

5. Final simplification 33.0%

   \[\leadsto -2 \]

Developer target: 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 2023293 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :herbie-target
  (- (* 4.0 (/ x z)) (+ 2.0 (* 4.0 (/ y z))))

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