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

Percentage Accurate: 99.8% → 100.0%

Time: 6.2s

Alternatives: 11

Speedup: 1.0×

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

Specification

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(4, 0.75 + \frac{x - z}{y}, 1\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma 4.0 (+ 0.75 (/ (- x z) y)) 1.0))

C

double code(double x, double y, double z) {
	return fma(4.0, (0.75 + ((x - z) / y)), 1.0);
}

Julia

function code(x, y, z)
	return fma(4.0, Float64(0.75 + Float64(Float64(x - z) / y)), 1.0)
end

Wolfram

code[x_, y_, z_] := N[(4.0 * N[(0.75 + N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. +-commutative 99.5%

      \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} + 1} \]

   2. associate-/l* 99.9%

      \[\leadsto \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} + 1 \]

   3. fma-define 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{\left(x + y \cdot 0.75\right) - z}{y}, 1\right)} \]

   4. associate--l+ 99.9%

      \[\leadsto \mathsf{fma}\left(4, \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y}, 1\right) \]

   5. +-commutative 99.9%

      \[\leadsto \mathsf{fma}\left(4, \frac{\color{blue}{\left(y \cdot 0.75 - z\right) + x}}{y}, 1\right) \]

   6. remove-double-neg 99.9%

      \[\leadsto \mathsf{fma}\left(4, \frac{\left(y \cdot 0.75 - z\right) + \color{blue}{\left(-\left(-x\right)\right)}}{y}, 1\right) \]

   7. sub-neg 99.9%

      \[\leadsto \mathsf{fma}\left(4, \frac{\color{blue}{\left(y \cdot 0.75 - z\right) - \left(-x\right)}}{y}, 1\right) \]

   8. associate--r+ 99.9%

      \[\leadsto \mathsf{fma}\left(4, \frac{\color{blue}{y \cdot 0.75 - \left(z + \left(-x\right)\right)}}{y}, 1\right) \]

   9. div-sub 99.9%

      \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{y \cdot 0.75}{y} - \frac{z + \left(-x\right)}{y}}, 1\right) \]

   10. sub-neg 99.9%

       \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{y \cdot 0.75}{y} + \left(-\frac{z + \left(-x\right)}{y}\right)}, 1\right) \]

   11. associate-*l/ 100.0%

       \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{y}{y} \cdot 0.75} + \left(-\frac{z + \left(-x\right)}{y}\right), 1\right) \]

   12. *-inverses 100.0%

       \[\leadsto \mathsf{fma}\left(4, \color{blue}{1} \cdot 0.75 + \left(-\frac{z + \left(-x\right)}{y}\right), 1\right) \]

   13. metadata-eval 100.0%

       \[\leadsto \mathsf{fma}\left(4, \color{blue}{0.75} + \left(-\frac{z + \left(-x\right)}{y}\right), 1\right) \]

   14. distribute-frac-neg2 100.0%

       \[\leadsto \mathsf{fma}\left(4, 0.75 + \color{blue}{\frac{z + \left(-x\right)}{-y}}, 1\right) \]

   15. remove-double-neg 100.0%

       \[\leadsto \mathsf{fma}\left(4, 0.75 + \frac{\color{blue}{\left(-\left(-z\right)\right)} + \left(-x\right)}{-y}, 1\right) \]

   16. distribute-neg-out 100.0%

       \[\leadsto \mathsf{fma}\left(4, 0.75 + \frac{\color{blue}{-\left(\left(-z\right) + x\right)}}{-y}, 1\right) \]

   17. +-commutative 100.0%

       \[\leadsto \mathsf{fma}\left(4, 0.75 + \frac{-\color{blue}{\left(x + \left(-z\right)\right)}}{-y}, 1\right) \]

   18. sub-neg 100.0%

       \[\leadsto \mathsf{fma}\left(4, 0.75 + \frac{-\color{blue}{\left(x - z\right)}}{-y}, 1\right) \]

   19. distribute-frac-neg 100.0%

       \[\leadsto \mathsf{fma}\left(4, 0.75 + \color{blue}{\left(-\frac{x - z}{-y}\right)}, 1\right) \]

   20. distribute-frac-neg2 100.0%

       \[\leadsto \mathsf{fma}\left(4, 0.75 + \color{blue}{\frac{x - z}{-\left(-y\right)}}, 1\right) \]

   21. remove-double-neg 100.0%

       \[\leadsto \mathsf{fma}\left(4, 0.75 + \frac{x - z}{\color{blue}{y}}, 1\right) \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(4, 0.75 + \frac{x - z}{y}, 1\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 53.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z \cdot -4}{y}\\ t_1 := \frac{4 \cdot x}{y}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+134}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+78}:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-120}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-192}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-273}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-183}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-138}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-80}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1650000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* z -4.0) y)) (t_1 (/ (* 4.0 x) y)))
   (if (<= x -2e+142)
     t_1
     (if (<= x -5.5e+134)
       4.0
       (if (<= x -8e+78)
         (* x (/ 4.0 y))
         (if (<= x -1.3e-120)
           t_0
           (if (<= x -1.7e-192)
             4.0
             (if (<= x -1.45e-273)
               t_0
               (if (<= x 1.1e-183)
                 4.0
                 (if (<= x 1.55e-138)
                   t_0
                   (if (<= x 3.8e-80)
                     4.0
                     (if (<= x 1650000000.0) t_0 t_1))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = (z * -4.0) / y;
	double t_1 = (4.0 * x) / y;
	double tmp;
	if (x <= -2e+142) {
		tmp = t_1;
	} else if (x <= -5.5e+134) {
		tmp = 4.0;
	} else if (x <= -8e+78) {
		tmp = x * (4.0 / y);
	} else if (x <= -1.3e-120) {
		tmp = t_0;
	} else if (x <= -1.7e-192) {
		tmp = 4.0;
	} else if (x <= -1.45e-273) {
		tmp = t_0;
	} else if (x <= 1.1e-183) {
		tmp = 4.0;
	} else if (x <= 1.55e-138) {
		tmp = t_0;
	} else if (x <= 3.8e-80) {
		tmp = 4.0;
	} else if (x <= 1650000000.0) {
		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 = (z * (-4.0d0)) / y
    t_1 = (4.0d0 * x) / y
    if (x <= (-2d+142)) then
        tmp = t_1
    else if (x <= (-5.5d+134)) then
        tmp = 4.0d0
    else if (x <= (-8d+78)) then
        tmp = x * (4.0d0 / y)
    else if (x <= (-1.3d-120)) then
        tmp = t_0
    else if (x <= (-1.7d-192)) then
        tmp = 4.0d0
    else if (x <= (-1.45d-273)) then
        tmp = t_0
    else if (x <= 1.1d-183) then
        tmp = 4.0d0
    else if (x <= 1.55d-138) then
        tmp = t_0
    else if (x <= 3.8d-80) then
        tmp = 4.0d0
    else if (x <= 1650000000.0d0) 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 = (z * -4.0) / y;
	double t_1 = (4.0 * x) / y;
	double tmp;
	if (x <= -2e+142) {
		tmp = t_1;
	} else if (x <= -5.5e+134) {
		tmp = 4.0;
	} else if (x <= -8e+78) {
		tmp = x * (4.0 / y);
	} else if (x <= -1.3e-120) {
		tmp = t_0;
	} else if (x <= -1.7e-192) {
		tmp = 4.0;
	} else if (x <= -1.45e-273) {
		tmp = t_0;
	} else if (x <= 1.1e-183) {
		tmp = 4.0;
	} else if (x <= 1.55e-138) {
		tmp = t_0;
	} else if (x <= 3.8e-80) {
		tmp = 4.0;
	} else if (x <= 1650000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * -4.0) / y
	t_1 = (4.0 * x) / y
	tmp = 0
	if x <= -2e+142:
		tmp = t_1
	elif x <= -5.5e+134:
		tmp = 4.0
	elif x <= -8e+78:
		tmp = x * (4.0 / y)
	elif x <= -1.3e-120:
		tmp = t_0
	elif x <= -1.7e-192:
		tmp = 4.0
	elif x <= -1.45e-273:
		tmp = t_0
	elif x <= 1.1e-183:
		tmp = 4.0
	elif x <= 1.55e-138:
		tmp = t_0
	elif x <= 3.8e-80:
		tmp = 4.0
	elif x <= 1650000000.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * -4.0) / y)
	t_1 = Float64(Float64(4.0 * x) / y)
	tmp = 0.0
	if (x <= -2e+142)
		tmp = t_1;
	elseif (x <= -5.5e+134)
		tmp = 4.0;
	elseif (x <= -8e+78)
		tmp = Float64(x * Float64(4.0 / y));
	elseif (x <= -1.3e-120)
		tmp = t_0;
	elseif (x <= -1.7e-192)
		tmp = 4.0;
	elseif (x <= -1.45e-273)
		tmp = t_0;
	elseif (x <= 1.1e-183)
		tmp = 4.0;
	elseif (x <= 1.55e-138)
		tmp = t_0;
	elseif (x <= 3.8e-80)
		tmp = 4.0;
	elseif (x <= 1650000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * -4.0) / y;
	t_1 = (4.0 * x) / y;
	tmp = 0.0;
	if (x <= -2e+142)
		tmp = t_1;
	elseif (x <= -5.5e+134)
		tmp = 4.0;
	elseif (x <= -8e+78)
		tmp = x * (4.0 / y);
	elseif (x <= -1.3e-120)
		tmp = t_0;
	elseif (x <= -1.7e-192)
		tmp = 4.0;
	elseif (x <= -1.45e-273)
		tmp = t_0;
	elseif (x <= 1.1e-183)
		tmp = 4.0;
	elseif (x <= 1.55e-138)
		tmp = t_0;
	elseif (x <= 3.8e-80)
		tmp = 4.0;
	elseif (x <= 1650000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -2e+142], t$95$1, If[LessEqual[x, -5.5e+134], 4.0, If[LessEqual[x, -8e+78], N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.3e-120], t$95$0, If[LessEqual[x, -1.7e-192], 4.0, If[LessEqual[x, -1.45e-273], t$95$0, If[LessEqual[x, 1.1e-183], 4.0, If[LessEqual[x, 1.55e-138], t$95$0, If[LessEqual[x, 3.8e-80], 4.0, If[LessEqual[x, 1650000000.0], t$95$0, t$95$1]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -2.0000000000000001e142 or 1.65e9 < x

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 99.9%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 72.9%

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

      1. *-commutative 72.9%

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

      2. associate-*l/ 72.9%

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

   7. Simplified 72.9%

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

   if -2.0000000000000001e142 < x < -5.4999999999999999e134 or -1.3000000000000001e-120 < x < -1.70000000000000001e-192 or -1.44999999999999993e-273 < x < 1.1e-183 or 1.5499999999999999e-138 < x < 3.79999999999999967e-80

   1. Initial program 98.6%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 99.9%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 67.3%

      \[\leadsto \color{blue}{4} \]

   if -5.4999999999999999e134 < x < -8.00000000000000007e78

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 100.0%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto 1 + 4 \cdot \color{blue}{\left(\left(0.75 + \frac{x}{y}\right) - \frac{z}{y}\right)} \]

   6. Taylor expanded in x around inf 88.2%

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

      1. *-commutative 88.2%

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

      2. associate-*l/ 88.2%

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

      3. associate-*r/ 88.2%

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

   8. Simplified 88.2%

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

   if -8.00000000000000007e78 < x < -1.3000000000000001e-120 or -1.70000000000000001e-192 < x < -1.44999999999999993e-273 or 1.1e-183 < x < 1.5499999999999999e-138 or 3.79999999999999967e-80 < x < 1.65e9

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 99.9%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 66.0%

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

      1. *-commutative 66.0%

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

      2. associate-*l/ 66.0%

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

   7. Simplified 66.0%

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

4. Final simplification 69.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+142}:\\ \;\;\;\;\frac{4 \cdot x}{y}\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+134}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq -8 \cdot 10^{+78}:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{-120}:\\ \;\;\;\;\frac{z \cdot -4}{y}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-192}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq -1.45 \cdot 10^{-273}:\\ \;\;\;\;\frac{z \cdot -4}{y}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-183}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-138}:\\ \;\;\;\;\frac{z \cdot -4}{y}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{-80}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1650000000:\\ \;\;\;\;\frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{-4}{y}\\ t_1 := \frac{4 \cdot x}{y}\\ \mathbf{if}\;x \leq -8.2 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -4 \cdot 10^{+134}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{+79}:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-192}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-269}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-185}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-139}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-81}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 980000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ -4.0 y))) (t_1 (/ (* 4.0 x) y)))
   (if (<= x -8.2e+141)
     t_1
     (if (<= x -4e+134)
       4.0
       (if (<= x -1.3e+79)
         (* x (/ 4.0 y))
         (if (<= x -6.2e-119)
           t_0
           (if (<= x -1.6e-192)
             4.0
             (if (<= x -1.9e-269)
               t_0
               (if (<= x 6e-185)
                 4.0
                 (if (<= x 1.3e-139)
                   t_0
                   (if (<= x 8.5e-81)
                     4.0
                     (if (<= x 980000.0) t_0 t_1))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double t_1 = (4.0 * x) / y;
	double tmp;
	if (x <= -8.2e+141) {
		tmp = t_1;
	} else if (x <= -4e+134) {
		tmp = 4.0;
	} else if (x <= -1.3e+79) {
		tmp = x * (4.0 / y);
	} else if (x <= -6.2e-119) {
		tmp = t_0;
	} else if (x <= -1.6e-192) {
		tmp = 4.0;
	} else if (x <= -1.9e-269) {
		tmp = t_0;
	} else if (x <= 6e-185) {
		tmp = 4.0;
	} else if (x <= 1.3e-139) {
		tmp = t_0;
	} else if (x <= 8.5e-81) {
		tmp = 4.0;
	} else if (x <= 980000.0) {
		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 = z * ((-4.0d0) / y)
    t_1 = (4.0d0 * x) / y
    if (x <= (-8.2d+141)) then
        tmp = t_1
    else if (x <= (-4d+134)) then
        tmp = 4.0d0
    else if (x <= (-1.3d+79)) then
        tmp = x * (4.0d0 / y)
    else if (x <= (-6.2d-119)) then
        tmp = t_0
    else if (x <= (-1.6d-192)) then
        tmp = 4.0d0
    else if (x <= (-1.9d-269)) then
        tmp = t_0
    else if (x <= 6d-185) then
        tmp = 4.0d0
    else if (x <= 1.3d-139) then
        tmp = t_0
    else if (x <= 8.5d-81) then
        tmp = 4.0d0
    else if (x <= 980000.0d0) 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 = z * (-4.0 / y);
	double t_1 = (4.0 * x) / y;
	double tmp;
	if (x <= -8.2e+141) {
		tmp = t_1;
	} else if (x <= -4e+134) {
		tmp = 4.0;
	} else if (x <= -1.3e+79) {
		tmp = x * (4.0 / y);
	} else if (x <= -6.2e-119) {
		tmp = t_0;
	} else if (x <= -1.6e-192) {
		tmp = 4.0;
	} else if (x <= -1.9e-269) {
		tmp = t_0;
	} else if (x <= 6e-185) {
		tmp = 4.0;
	} else if (x <= 1.3e-139) {
		tmp = t_0;
	} else if (x <= 8.5e-81) {
		tmp = 4.0;
	} else if (x <= 980000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-4.0 / y)
	t_1 = (4.0 * x) / y
	tmp = 0
	if x <= -8.2e+141:
		tmp = t_1
	elif x <= -4e+134:
		tmp = 4.0
	elif x <= -1.3e+79:
		tmp = x * (4.0 / y)
	elif x <= -6.2e-119:
		tmp = t_0
	elif x <= -1.6e-192:
		tmp = 4.0
	elif x <= -1.9e-269:
		tmp = t_0
	elif x <= 6e-185:
		tmp = 4.0
	elif x <= 1.3e-139:
		tmp = t_0
	elif x <= 8.5e-81:
		tmp = 4.0
	elif x <= 980000.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-4.0 / y))
	t_1 = Float64(Float64(4.0 * x) / y)
	tmp = 0.0
	if (x <= -8.2e+141)
		tmp = t_1;
	elseif (x <= -4e+134)
		tmp = 4.0;
	elseif (x <= -1.3e+79)
		tmp = Float64(x * Float64(4.0 / y));
	elseif (x <= -6.2e-119)
		tmp = t_0;
	elseif (x <= -1.6e-192)
		tmp = 4.0;
	elseif (x <= -1.9e-269)
		tmp = t_0;
	elseif (x <= 6e-185)
		tmp = 4.0;
	elseif (x <= 1.3e-139)
		tmp = t_0;
	elseif (x <= 8.5e-81)
		tmp = 4.0;
	elseif (x <= 980000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-4.0 / y);
	t_1 = (4.0 * x) / y;
	tmp = 0.0;
	if (x <= -8.2e+141)
		tmp = t_1;
	elseif (x <= -4e+134)
		tmp = 4.0;
	elseif (x <= -1.3e+79)
		tmp = x * (4.0 / y);
	elseif (x <= -6.2e-119)
		tmp = t_0;
	elseif (x <= -1.6e-192)
		tmp = 4.0;
	elseif (x <= -1.9e-269)
		tmp = t_0;
	elseif (x <= 6e-185)
		tmp = 4.0;
	elseif (x <= 1.3e-139)
		tmp = t_0;
	elseif (x <= 8.5e-81)
		tmp = 4.0;
	elseif (x <= 980000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -8.2e+141], t$95$1, If[LessEqual[x, -4e+134], 4.0, If[LessEqual[x, -1.3e+79], N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -6.2e-119], t$95$0, If[LessEqual[x, -1.6e-192], 4.0, If[LessEqual[x, -1.9e-269], t$95$0, If[LessEqual[x, 6e-185], 4.0, If[LessEqual[x, 1.3e-139], t$95$0, If[LessEqual[x, 8.5e-81], 4.0, If[LessEqual[x, 980000.0], t$95$0, t$95$1]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -8.20000000000000044e141 or 9.8e5 < x

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 99.9%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 72.9%

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

      1. *-commutative 72.9%

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

      2. associate-*l/ 72.9%

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

   7. Simplified 72.9%

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

   if -8.20000000000000044e141 < x < -3.99999999999999969e134 or -6.19999999999999956e-119 < x < -1.6000000000000001e-192 or -1.9000000000000001e-269 < x < 6.00000000000000061e-185 or 1.2999999999999999e-139 < x < 8.5000000000000001e-81

   1. Initial program 98.6%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 99.9%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 67.3%

      \[\leadsto \color{blue}{4} \]

   if -3.99999999999999969e134 < x < -1.30000000000000007e79

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 100.0%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto 1 + 4 \cdot \color{blue}{\left(\left(0.75 + \frac{x}{y}\right) - \frac{z}{y}\right)} \]

   6. Taylor expanded in x around inf 88.2%

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

      1. *-commutative 88.2%

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

      2. associate-*l/ 88.2%

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

      3. associate-*r/ 88.2%

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

   8. Simplified 88.2%

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

   if -1.30000000000000007e79 < x < -6.19999999999999956e-119 or -1.6000000000000001e-192 < x < -1.9000000000000001e-269 or 6.00000000000000061e-185 < x < 1.2999999999999999e-139 or 8.5000000000000001e-81 < x < 9.8e5

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 99.9%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto 1 + 4 \cdot \color{blue}{\left(\left(0.75 + \frac{x}{y}\right) - \frac{z}{y}\right)} \]

   6. Taylor expanded in z around inf 66.0%

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

      1. metadata-eval 66.0%

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

      2. distribute-lft-neg-in 66.0%

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

      3. *-lft-identity 66.0%

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

      4. associate-*l/ 65.7%

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

      5. associate-*l* 65.7%

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

      6. *-commutative 65.7%

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

      7. distribute-rgt-neg-in 65.7%

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

      8. associate-*r/ 65.7%

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

      9. metadata-eval 65.7%

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

      10. distribute-neg-frac 65.7%

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

      11. metadata-eval 65.7%

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

   8. Simplified 65.7%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.2 \cdot 10^{+141}:\\ \;\;\;\;\frac{4 \cdot x}{y}\\ \mathbf{elif}\;x \leq -4 \cdot 10^{+134}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq -1.3 \cdot 10^{+79}:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-119}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;x \leq -1.6 \cdot 10^{-192}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq -1.9 \cdot 10^{-269}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-185}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-139}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-81}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 980000:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{-4}{y}\\ t_1 := x \cdot \frac{4}{y}\\ \mathbf{if}\;x \leq -7.6 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -5.5 \cdot 10^{+134}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.95 \cdot 10^{-119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -1.75 \cdot 10^{-192}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-274}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-183}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-139}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.36 \cdot 10^{-80}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 65000000:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ -4.0 y))) (t_1 (* x (/ 4.0 y))))
   (if (<= x -7.6e+141)
     t_1
     (if (<= x -5.5e+134)
       4.0
       (if (<= x -1.7e+79)
         t_1
         (if (<= x -1.95e-119)
           t_0
           (if (<= x -1.75e-192)
             4.0
             (if (<= x -8e-274)
               t_0
               (if (<= x 1.3e-183)
                 4.0
                 (if (<= x 1.35e-139)
                   t_0
                   (if (<= x 1.36e-80)
                     4.0
                     (if (<= x 65000000.0) t_0 t_1))))))))))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double t_1 = x * (4.0 / y);
	double tmp;
	if (x <= -7.6e+141) {
		tmp = t_1;
	} else if (x <= -5.5e+134) {
		tmp = 4.0;
	} else if (x <= -1.7e+79) {
		tmp = t_1;
	} else if (x <= -1.95e-119) {
		tmp = t_0;
	} else if (x <= -1.75e-192) {
		tmp = 4.0;
	} else if (x <= -8e-274) {
		tmp = t_0;
	} else if (x <= 1.3e-183) {
		tmp = 4.0;
	} else if (x <= 1.35e-139) {
		tmp = t_0;
	} else if (x <= 1.36e-80) {
		tmp = 4.0;
	} else if (x <= 65000000.0) {
		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 = z * ((-4.0d0) / y)
    t_1 = x * (4.0d0 / y)
    if (x <= (-7.6d+141)) then
        tmp = t_1
    else if (x <= (-5.5d+134)) then
        tmp = 4.0d0
    else if (x <= (-1.7d+79)) then
        tmp = t_1
    else if (x <= (-1.95d-119)) then
        tmp = t_0
    else if (x <= (-1.75d-192)) then
        tmp = 4.0d0
    else if (x <= (-8d-274)) then
        tmp = t_0
    else if (x <= 1.3d-183) then
        tmp = 4.0d0
    else if (x <= 1.35d-139) then
        tmp = t_0
    else if (x <= 1.36d-80) then
        tmp = 4.0d0
    else if (x <= 65000000.0d0) 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 = z * (-4.0 / y);
	double t_1 = x * (4.0 / y);
	double tmp;
	if (x <= -7.6e+141) {
		tmp = t_1;
	} else if (x <= -5.5e+134) {
		tmp = 4.0;
	} else if (x <= -1.7e+79) {
		tmp = t_1;
	} else if (x <= -1.95e-119) {
		tmp = t_0;
	} else if (x <= -1.75e-192) {
		tmp = 4.0;
	} else if (x <= -8e-274) {
		tmp = t_0;
	} else if (x <= 1.3e-183) {
		tmp = 4.0;
	} else if (x <= 1.35e-139) {
		tmp = t_0;
	} else if (x <= 1.36e-80) {
		tmp = 4.0;
	} else if (x <= 65000000.0) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-4.0 / y)
	t_1 = x * (4.0 / y)
	tmp = 0
	if x <= -7.6e+141:
		tmp = t_1
	elif x <= -5.5e+134:
		tmp = 4.0
	elif x <= -1.7e+79:
		tmp = t_1
	elif x <= -1.95e-119:
		tmp = t_0
	elif x <= -1.75e-192:
		tmp = 4.0
	elif x <= -8e-274:
		tmp = t_0
	elif x <= 1.3e-183:
		tmp = 4.0
	elif x <= 1.35e-139:
		tmp = t_0
	elif x <= 1.36e-80:
		tmp = 4.0
	elif x <= 65000000.0:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-4.0 / y))
	t_1 = Float64(x * Float64(4.0 / y))
	tmp = 0.0
	if (x <= -7.6e+141)
		tmp = t_1;
	elseif (x <= -5.5e+134)
		tmp = 4.0;
	elseif (x <= -1.7e+79)
		tmp = t_1;
	elseif (x <= -1.95e-119)
		tmp = t_0;
	elseif (x <= -1.75e-192)
		tmp = 4.0;
	elseif (x <= -8e-274)
		tmp = t_0;
	elseif (x <= 1.3e-183)
		tmp = 4.0;
	elseif (x <= 1.35e-139)
		tmp = t_0;
	elseif (x <= 1.36e-80)
		tmp = 4.0;
	elseif (x <= 65000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-4.0 / y);
	t_1 = x * (4.0 / y);
	tmp = 0.0;
	if (x <= -7.6e+141)
		tmp = t_1;
	elseif (x <= -5.5e+134)
		tmp = 4.0;
	elseif (x <= -1.7e+79)
		tmp = t_1;
	elseif (x <= -1.95e-119)
		tmp = t_0;
	elseif (x <= -1.75e-192)
		tmp = 4.0;
	elseif (x <= -8e-274)
		tmp = t_0;
	elseif (x <= 1.3e-183)
		tmp = 4.0;
	elseif (x <= 1.35e-139)
		tmp = t_0;
	elseif (x <= 1.36e-80)
		tmp = 4.0;
	elseif (x <= 65000000.0)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -7.6e+141], t$95$1, If[LessEqual[x, -5.5e+134], 4.0, If[LessEqual[x, -1.7e+79], t$95$1, If[LessEqual[x, -1.95e-119], t$95$0, If[LessEqual[x, -1.75e-192], 4.0, If[LessEqual[x, -8e-274], t$95$0, If[LessEqual[x, 1.3e-183], 4.0, If[LessEqual[x, 1.35e-139], t$95$0, If[LessEqual[x, 1.36e-80], 4.0, If[LessEqual[x, 65000000.0], t$95$0, t$95$1]]]]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -7.59999999999999952e141 or -5.4999999999999999e134 < x < -1.70000000000000016e79 or 6.5e7 < x

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 99.9%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 95.9%

      \[\leadsto 1 + 4 \cdot \color{blue}{\left(\left(0.75 + \frac{x}{y}\right) - \frac{z}{y}\right)} \]

   6. Taylor expanded in x around inf 74.1%

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

      1. *-commutative 74.1%

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

      2. associate-*l/ 74.1%

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

      3. associate-*r/ 74.0%

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

   8. Simplified 74.0%

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

   if -7.59999999999999952e141 < x < -5.4999999999999999e134 or -1.94999999999999995e-119 < x < -1.75000000000000007e-192 or -7.99999999999999973e-274 < x < 1.2999999999999999e-183 or 1.3499999999999999e-139 < x < 1.3599999999999999e-80

   1. Initial program 98.6%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 99.9%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 67.3%

      \[\leadsto \color{blue}{4} \]

   if -1.70000000000000016e79 < x < -1.94999999999999995e-119 or -1.75000000000000007e-192 < x < -7.99999999999999973e-274 or 1.2999999999999999e-183 < x < 1.3499999999999999e-139 or 1.3599999999999999e-80 < x < 6.5e7

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 99.9%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto 1 + 4 \cdot \color{blue}{\left(\left(0.75 + \frac{x}{y}\right) - \frac{z}{y}\right)} \]

   6. Taylor expanded in z around inf 66.0%

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

      1. metadata-eval 66.0%

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

      2. distribute-lft-neg-in 66.0%

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

      3. *-lft-identity 66.0%

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

      4. associate-*l/ 65.7%

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

      5. associate-*l* 65.7%

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

      6. *-commutative 65.7%

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

      7. distribute-rgt-neg-in 65.7%

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

      8. associate-*r/ 65.7%

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

      9. metadata-eval 65.7%

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

      10. distribute-neg-frac 65.7%

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

      11. metadata-eval 65.7%

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

   8. Simplified 65.7%

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

Alternative 5: 52.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+141} \lor \neg \left(x \leq -6 \cdot 10^{+133} \lor \neg \left(x \leq -8 \cdot 10^{+71}\right) \land x \leq 6.5 \cdot 10^{-45}\right):\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -7.5e+141)
         (not (or (<= x -6e+133) (and (not (<= x -8e+71)) (<= x 6.5e-45)))))
   (* x (/ 4.0 y))
   4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.5e+141) || !((x <= -6e+133) || (!(x <= -8e+71) && (x <= 6.5e-45)))) {
		tmp = x * (4.0 / y);
	} else {
		tmp = 4.0;
	}
	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 <= (-7.5d+141)) .or. (.not. (x <= (-6d+133)) .or. (.not. (x <= (-8d+71))) .and. (x <= 6.5d-45))) then
        tmp = x * (4.0d0 / y)
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -7.5e+141) || !((x <= -6e+133) || (!(x <= -8e+71) && (x <= 6.5e-45)))) {
		tmp = x * (4.0 / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -7.5e+141) or not ((x <= -6e+133) or (not (x <= -8e+71) and (x <= 6.5e-45))):
		tmp = x * (4.0 / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -7.5e+141) || !((x <= -6e+133) || (!(x <= -8e+71) && (x <= 6.5e-45))))
		tmp = Float64(x * Float64(4.0 / y));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -7.5e+141) || ~(((x <= -6e+133) || (~((x <= -8e+71)) && (x <= 6.5e-45)))))
		tmp = x * (4.0 / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -7.5e+141], N[Not[Or[LessEqual[x, -6e+133], And[N[Not[LessEqual[x, -8e+71]], $MachinePrecision], LessEqual[x, 6.5e-45]]]], $MachinePrecision]], N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if x < -7.49999999999999937e141 or -6.00000000000000013e133 < x < -8.0000000000000003e71 or 6.4999999999999995e-45 < x

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 99.9%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 96.4%

      \[\leadsto 1 + 4 \cdot \color{blue}{\left(\left(0.75 + \frac{x}{y}\right) - \frac{z}{y}\right)} \]

   6. Taylor expanded in x around inf 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-*l/ 69.8%

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

      3. associate-*r/ 69.6%

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

   8. Simplified 69.6%

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

   if -7.49999999999999937e141 < x < -6.00000000000000013e133 or -8.0000000000000003e71 < x < 6.4999999999999995e-45

   1. Initial program 99.2%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 99.9%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 49.1%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -7.5 \cdot 10^{+141} \lor \neg \left(x \leq -6 \cdot 10^{+133} \lor \neg \left(x \leq -8 \cdot 10^{+71}\right) \land x \leq 6.5 \cdot 10^{-45}\right):\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+40} \lor \neg \left(x \leq 320000000\right):\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -8.5e+40) (not (<= x 320000000.0)))
   (+ 4.0 (/ (* 4.0 x) y))
   (+ 4.0 (/ (* z -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.5e+40) || !(x <= 320000000.0)) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 + ((z * -4.0) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-8.5d+40)) .or. (.not. (x <= 320000000.0d0))) then
        tmp = 4.0d0 + ((4.0d0 * x) / y)
    else
        tmp = 4.0d0 + ((z * (-4.0d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -8.5e+40) || !(x <= 320000000.0)) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 + ((z * -4.0) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -8.5e+40) or not (x <= 320000000.0):
		tmp = 4.0 + ((4.0 * x) / y)
	else:
		tmp = 4.0 + ((z * -4.0) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -8.5e+40) || !(x <= 320000000.0))
		tmp = Float64(4.0 + Float64(Float64(4.0 * x) / y));
	else
		tmp = Float64(4.0 + Float64(Float64(z * -4.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -8.5e+40) || ~((x <= 320000000.0)))
		tmp = 4.0 + ((4.0 * x) / y);
	else
		tmp = 4.0 + ((z * -4.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -8.5e+40], N[Not[LessEqual[x, 320000000.0]], $MachinePrecision]], N[(4.0 + N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.49999999999999996e40 or 3.2e8 < x

   1. Initial program 99.1%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} + 1} \]

      2. associate-/l* 99.9%

         \[\leadsto \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} + 1 \]

      3. fma-define 99.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{\left(x + y \cdot 0.75\right) - z}{y}, 1\right)} \]

      4. associate--l+ 99.9%

         \[\leadsto \mathsf{fma}\left(4, \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y}, 1\right) \]

      5. +-commutative 99.9%

         \[\leadsto \mathsf{fma}\left(4, \frac{\color{blue}{\left(y \cdot 0.75 - z\right) + x}}{y}, 1\right) \]

      6. remove-double-neg 99.9%

         \[\leadsto \mathsf{fma}\left(4, \frac{\left(y \cdot 0.75 - z\right) + \color{blue}{\left(-\left(-x\right)\right)}}{y}, 1\right) \]

      7. sub-neg 99.9%

         \[\leadsto \mathsf{fma}\left(4, \frac{\color{blue}{\left(y \cdot 0.75 - z\right) - \left(-x\right)}}{y}, 1\right) \]

      8. associate--r+ 99.9%

         \[\leadsto \mathsf{fma}\left(4, \frac{\color{blue}{y \cdot 0.75 - \left(z + \left(-x\right)\right)}}{y}, 1\right) \]

      9. div-sub 100.0%

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{y \cdot 0.75}{y} - \frac{z + \left(-x\right)}{y}}, 1\right) \]

      10. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{y \cdot 0.75}{y} + \left(-\frac{z + \left(-x\right)}{y}\right)}, 1\right) \]

      11. associate-*l/ 100.0%

          \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{y}{y} \cdot 0.75} + \left(-\frac{z + \left(-x\right)}{y}\right), 1\right) \]

      12. *-inverses 100.0%

          \[\leadsto \mathsf{fma}\left(4, \color{blue}{1} \cdot 0.75 + \left(-\frac{z + \left(-x\right)}{y}\right), 1\right) \]

      13. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(4, \color{blue}{0.75} + \left(-\frac{z + \left(-x\right)}{y}\right), 1\right) \]

      14. distribute-frac-neg2 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \color{blue}{\frac{z + \left(-x\right)}{-y}}, 1\right) \]

      15. remove-double-neg 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \frac{\color{blue}{\left(-\left(-z\right)\right)} + \left(-x\right)}{-y}, 1\right) \]

      16. distribute-neg-out 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \frac{\color{blue}{-\left(\left(-z\right) + x\right)}}{-y}, 1\right) \]

      17. +-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \frac{-\color{blue}{\left(x + \left(-z\right)\right)}}{-y}, 1\right) \]

      18. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \frac{-\color{blue}{\left(x - z\right)}}{-y}, 1\right) \]

      19. distribute-frac-neg 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \color{blue}{\left(-\frac{x - z}{-y}\right)}, 1\right) \]

      20. distribute-frac-neg2 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \color{blue}{\frac{x - z}{-\left(-y\right)}}, 1\right) \]

      21. remove-double-neg 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \frac{x - z}{\color{blue}{y}}, 1\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, 0.75 + \frac{x - z}{y}, 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 88.5%

      \[\leadsto \color{blue}{1 + 4 \cdot \left(0.75 + \frac{x}{y}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 88.5%

         \[\leadsto 1 + \color{blue}{\left(4 \cdot 0.75 + 4 \cdot \frac{x}{y}\right)} \]

      2. metadata-eval 88.5%

         \[\leadsto 1 + \left(\color{blue}{3} + 4 \cdot \frac{x}{y}\right) \]

      3. associate-+r+ 88.5%

         \[\leadsto \color{blue}{\left(1 + 3\right) + 4 \cdot \frac{x}{y}} \]

      4. metadata-eval 88.5%

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

      5. *-commutative 88.5%

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

      6. associate-*l/ 88.5%

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

   7. Simplified 88.5%

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

   if -8.49999999999999996e40 < x < 3.2e8

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 99.9%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 95.1%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{0.75 \cdot y - z}{y}} \]
   6. Step-by-step derivation

      1. div-sub 95.1%

         \[\leadsto 1 + 4 \cdot \color{blue}{\left(\frac{0.75 \cdot y}{y} - \frac{z}{y}\right)} \]

      2. associate-/l* 95.2%

         \[\leadsto 1 + 4 \cdot \left(\color{blue}{0.75 \cdot \frac{y}{y}} - \frac{z}{y}\right) \]

      3. *-inverses 95.2%

         \[\leadsto 1 + 4 \cdot \left(0.75 \cdot \color{blue}{1} - \frac{z}{y}\right) \]

      4. metadata-eval 95.2%

         \[\leadsto 1 + 4 \cdot \left(\color{blue}{0.75} - \frac{z}{y}\right) \]

      5. sub-neg 95.2%

         \[\leadsto 1 + 4 \cdot \color{blue}{\left(0.75 + \left(-\frac{z}{y}\right)\right)} \]

      6. distribute-rgt-in 95.2%

         \[\leadsto 1 + \color{blue}{\left(0.75 \cdot 4 + \left(-\frac{z}{y}\right) \cdot 4\right)} \]

      7. metadata-eval 95.2%

         \[\leadsto 1 + \left(\color{blue}{3} + \left(-\frac{z}{y}\right) \cdot 4\right) \]

      8. associate-+r+ 95.2%

         \[\leadsto \color{blue}{\left(1 + 3\right) + \left(-\frac{z}{y}\right) \cdot 4} \]

      9. metadata-eval 95.2%

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

      10. neg-mul-1 95.2%

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

      11. *-commutative 95.2%

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

      12. associate-*l* 95.2%

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

      13. metadata-eval 95.2%

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

      14. associate-*l/ 95.2%

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

   7. Simplified 95.2%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+40} \lor \neg \left(x \leq 320000000\right):\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 85.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5400000 \lor \neg \left(y \leq 9000000000\right):\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5400000.0) (not (<= y 9000000000.0)))
   (+ 4.0 (/ (* 4.0 x) y))
   (* 4.0 (/ (- x z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5400000.0) || !(y <= 9000000000.0)) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 * ((x - z) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-5400000.0d0)) .or. (.not. (y <= 9000000000.0d0))) then
        tmp = 4.0d0 + ((4.0d0 * x) / y)
    else
        tmp = 4.0d0 * ((x - z) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5400000.0) || !(y <= 9000000000.0)) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 * ((x - z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -5400000.0) or not (y <= 9000000000.0):
		tmp = 4.0 + ((4.0 * x) / y)
	else:
		tmp = 4.0 * ((x - z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5400000.0) || !(y <= 9000000000.0))
		tmp = Float64(4.0 + Float64(Float64(4.0 * x) / y));
	else
		tmp = Float64(4.0 * Float64(Float64(x - z) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -5400000.0) || ~((y <= 9000000000.0)))
		tmp = 4.0 + ((4.0 * x) / y);
	else
		tmp = 4.0 * ((x - z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5400000.0], N[Not[LessEqual[y, 9000000000.0]], $MachinePrecision]], N[(4.0 + N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4e6 or 9e9 < y

   1. Initial program 99.1%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. +-commutative 99.1%

         \[\leadsto \color{blue}{\frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} + 1} \]

      2. associate-/l* 99.9%

         \[\leadsto \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} + 1 \]

      3. fma-define 99.9%

         \[\leadsto \color{blue}{\mathsf{fma}\left(4, \frac{\left(x + y \cdot 0.75\right) - z}{y}, 1\right)} \]

      4. associate--l+ 99.9%

         \[\leadsto \mathsf{fma}\left(4, \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y}, 1\right) \]

      5. +-commutative 99.9%

         \[\leadsto \mathsf{fma}\left(4, \frac{\color{blue}{\left(y \cdot 0.75 - z\right) + x}}{y}, 1\right) \]

      6. remove-double-neg 99.9%

         \[\leadsto \mathsf{fma}\left(4, \frac{\left(y \cdot 0.75 - z\right) + \color{blue}{\left(-\left(-x\right)\right)}}{y}, 1\right) \]

      7. sub-neg 99.9%

         \[\leadsto \mathsf{fma}\left(4, \frac{\color{blue}{\left(y \cdot 0.75 - z\right) - \left(-x\right)}}{y}, 1\right) \]

      8. associate--r+ 99.9%

         \[\leadsto \mathsf{fma}\left(4, \frac{\color{blue}{y \cdot 0.75 - \left(z + \left(-x\right)\right)}}{y}, 1\right) \]

      9. div-sub 99.9%

         \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{y \cdot 0.75}{y} - \frac{z + \left(-x\right)}{y}}, 1\right) \]

      10. sub-neg 99.9%

          \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{y \cdot 0.75}{y} + \left(-\frac{z + \left(-x\right)}{y}\right)}, 1\right) \]

      11. associate-*l/ 100.0%

          \[\leadsto \mathsf{fma}\left(4, \color{blue}{\frac{y}{y} \cdot 0.75} + \left(-\frac{z + \left(-x\right)}{y}\right), 1\right) \]

      12. *-inverses 100.0%

          \[\leadsto \mathsf{fma}\left(4, \color{blue}{1} \cdot 0.75 + \left(-\frac{z + \left(-x\right)}{y}\right), 1\right) \]

      13. metadata-eval 100.0%

          \[\leadsto \mathsf{fma}\left(4, \color{blue}{0.75} + \left(-\frac{z + \left(-x\right)}{y}\right), 1\right) \]

      14. distribute-frac-neg2 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \color{blue}{\frac{z + \left(-x\right)}{-y}}, 1\right) \]

      15. remove-double-neg 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \frac{\color{blue}{\left(-\left(-z\right)\right)} + \left(-x\right)}{-y}, 1\right) \]

      16. distribute-neg-out 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \frac{\color{blue}{-\left(\left(-z\right) + x\right)}}{-y}, 1\right) \]

      17. +-commutative 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \frac{-\color{blue}{\left(x + \left(-z\right)\right)}}{-y}, 1\right) \]

      18. sub-neg 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \frac{-\color{blue}{\left(x - z\right)}}{-y}, 1\right) \]

      19. distribute-frac-neg 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \color{blue}{\left(-\frac{x - z}{-y}\right)}, 1\right) \]

      20. distribute-frac-neg2 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \color{blue}{\frac{x - z}{-\left(-y\right)}}, 1\right) \]

      21. remove-double-neg 100.0%

          \[\leadsto \mathsf{fma}\left(4, 0.75 + \frac{x - z}{\color{blue}{y}}, 1\right) \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(4, 0.75 + \frac{x - z}{y}, 1\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 83.6%

      \[\leadsto \color{blue}{1 + 4 \cdot \left(0.75 + \frac{x}{y}\right)} \]
   6. Step-by-step derivation

      1. distribute-lft-in 83.6%

         \[\leadsto 1 + \color{blue}{\left(4 \cdot 0.75 + 4 \cdot \frac{x}{y}\right)} \]

      2. metadata-eval 83.6%

         \[\leadsto 1 + \left(\color{blue}{3} + 4 \cdot \frac{x}{y}\right) \]

      3. associate-+r+ 83.6%

         \[\leadsto \color{blue}{\left(1 + 3\right) + 4 \cdot \frac{x}{y}} \]

      4. metadata-eval 83.6%

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

      5. *-commutative 83.6%

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

      6. associate-*l/ 83.6%

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

   7. Simplified 83.6%

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

   if -5.4e6 < y < 9e9

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 99.9%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 93.5%

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

      1. *-commutative 93.5%

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

   7. Simplified 93.5%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5400000 \lor \neg \left(y \leq 9000000000\right):\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+95}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+76}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3e+95) 4.0 (if (<= y 1.65e+76) (* 4.0 (/ (- x z) y)) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3e+95) {
		tmp = 4.0;
	} else if (y <= 1.65e+76) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 4.0;
	}
	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 (y <= (-3d+95)) then
        tmp = 4.0d0
    else if (y <= 1.65d+76) then
        tmp = 4.0d0 * ((x - z) / y)
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3e+95) {
		tmp = 4.0;
	} else if (y <= 1.65e+76) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3e+95:
		tmp = 4.0
	elif y <= 1.65e+76:
		tmp = 4.0 * ((x - z) / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3e+95)
		tmp = 4.0;
	elseif (y <= 1.65e+76)
		tmp = Float64(4.0 * Float64(Float64(x - z) / y));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3e+95)
		tmp = 4.0;
	elseif (y <= 1.65e+76)
		tmp = 4.0 * ((x - z) / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3e+95], 4.0, If[LessEqual[y, 1.65e+76], N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.99999999999999991e95 or 1.65e76 < y

   1. Initial program 98.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 99.9%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around inf 69.8%

      \[\leadsto \color{blue}{4} \]

   if -2.99999999999999991e95 < y < 1.65e76

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.9%

         \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

      2. associate--l+ 99.9%

         \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 87.6%

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

      1. *-commutative 87.6%

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

   7. Simplified 87.6%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+95}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{+76}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + 4 \cdot \frac{x + \left(0.75 \cdot y - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (* 4.0 (/ (+ x (- (* 0.75 y) z)) y))))

C

double code(double x, double y, double z) {
	return 1.0 + (4.0 * ((x + ((0.75 * y) - z)) / y));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + (4.0d0 * ((x + ((0.75d0 * y) - z)) / y))
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + (4.0 * ((x + ((0.75 * y) - z)) / y));
}

Python

def code(x, y, z):
	return 1.0 + (4.0 * ((x + ((0.75 * y) - z)) / y))

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(4.0 * Float64(Float64(x + Float64(Float64(0.75 * y) - z)) / y)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + (4.0 * ((x + ((0.75 * y) - z)) / y));
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(4.0 * N[(N[(x + N[(N[(0.75 * y), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. associate-/l* 99.9%

      \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

   2. associate--l+ 99.9%

      \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
4. Add Preprocessing

5. Final simplification 99.9%

   \[\leadsto 1 + 4 \cdot \frac{x + \left(0.75 \cdot y - z\right)}{y} \]
6. Add Preprocessing

Alternative 10: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + 4 \cdot \left(\left(0.75 + \frac{x}{y}\right) - \frac{z}{y}\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (* 4.0 (- (+ 0.75 (/ x y)) (/ z y)))))

C

double code(double x, double y, double z) {
	return 1.0 + (4.0 * ((0.75 + (x / y)) - (z / y)));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + (4.0d0 * ((0.75d0 + (x / y)) - (z / y)))
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + (4.0 * ((0.75 + (x / y)) - (z / y)));
}

Python

def code(x, y, z):
	return 1.0 + (4.0 * ((0.75 + (x / y)) - (z / y)))

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(4.0 * Float64(Float64(0.75 + Float64(x / y)) - Float64(z / y))))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + (4.0 * ((0.75 + (x / y)) - (z / y)));
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(4.0 * N[(N[(0.75 + N[(x / y), $MachinePrecision]), $MachinePrecision] - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. associate-/l* 99.9%

      \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

   2. associate--l+ 99.9%

      \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 98.4%

   \[\leadsto 1 + 4 \cdot \color{blue}{\left(\left(0.75 + \frac{x}{y}\right) - \frac{z}{y}\right)} \]
6. Add Preprocessing

Alternative 11: 34.9% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 4 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 4.0)

C

double code(double x, double y, double z) {
	return 4.0;
}

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
end function

Java

public static double code(double x, double y, double z) {
	return 4.0;
}

Python

def code(x, y, z):
	return 4.0

Julia

function code(x, y, z)
	return 4.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 4.0;
end

Wolfram

code[x_, y_, z_] := 4.0

Derivation

1. Initial program 99.5%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. associate-/l* 99.9%

      \[\leadsto 1 + \color{blue}{4 \cdot \frac{\left(x + y \cdot 0.75\right) - z}{y}} \]

   2. associate--l+ 99.9%

      \[\leadsto 1 + 4 \cdot \frac{\color{blue}{x + \left(y \cdot 0.75 - z\right)}}{y} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{1 + 4 \cdot \frac{x + \left(y \cdot 0.75 - z\right)}{y}} \]
4. Add Preprocessing

5. Taylor expanded in y around inf 34.6%

   \[\leadsto \color{blue}{4} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024101 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))