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

Percentage Accurate: 99.8% → 100.0%

Time: 8.7s

Alternatives: 11

Speedup: 1.4×

• 21.1% 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 99.6%

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

   1. +-commutative 99.6%

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

   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: 54.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z \cdot -4}{y}\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-245}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+91}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* z -4.0) y)))
   (if (<= z -5.8e+42)
     t_0
     (if (<= z -1.12e-245) (* 4.0 (/ x y)) (if (<= z 6.5e+91) 4.0 t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (z * -4.0) / y;
	double tmp;
	if (z <= -5.8e+42) {
		tmp = t_0;
	} else if (z <= -1.12e-245) {
		tmp = 4.0 * (x / y);
	} else if (z <= 6.5e+91) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z * (-4.0d0)) / y
    if (z <= (-5.8d+42)) then
        tmp = t_0
    else if (z <= (-1.12d-245)) then
        tmp = 4.0d0 * (x / y)
    else if (z <= 6.5d+91) then
        tmp = 4.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z * -4.0) / y;
	double tmp;
	if (z <= -5.8e+42) {
		tmp = t_0;
	} else if (z <= -1.12e-245) {
		tmp = 4.0 * (x / y);
	} else if (z <= 6.5e+91) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z * -4.0) / y
	tmp = 0
	if z <= -5.8e+42:
		tmp = t_0
	elif z <= -1.12e-245:
		tmp = 4.0 * (x / y)
	elif z <= 6.5e+91:
		tmp = 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z * -4.0) / y)
	tmp = 0.0
	if (z <= -5.8e+42)
		tmp = t_0;
	elseif (z <= -1.12e-245)
		tmp = Float64(4.0 * Float64(x / y));
	elseif (z <= 6.5e+91)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z * -4.0) / y;
	tmp = 0.0;
	if (z <= -5.8e+42)
		tmp = t_0;
	elseif (z <= -1.12e-245)
		tmp = 4.0 * (x / y);
	elseif (z <= 6.5e+91)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[z, -5.8e+42], t$95$0, If[LessEqual[z, -1.12e-245], N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.5e+91], 4.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.79999999999999961e42 or 6.4999999999999997e91 < z

   1. Initial program 99.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   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 z around inf 75.2%

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

      1. *-commutative 75.2%

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

      2. associate-*l/ 75.2%

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

   7. Simplified 75.2%

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

   if -5.79999999999999961e42 < z < -1.12e-245

   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 66.1%

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

      1. *-commutative 66.1%

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

   7. Simplified 66.1%

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

   if -1.12e-245 < z < 6.4999999999999997e91

   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 inf 52.4%

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

4. Final simplification 64.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+42}:\\ \;\;\;\;\frac{z \cdot -4}{y}\\ \mathbf{elif}\;z \leq -1.12 \cdot 10^{-245}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+91}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot -4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{-4}{y}\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+43}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-243}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+91}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ -4.0 y))))
   (if (<= z -1.2e+43)
     t_0
     (if (<= z -1.8e-243) (* 4.0 (/ x y)) (if (<= z 6.5e+91) 4.0 t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double tmp;
	if (z <= -1.2e+43) {
		tmp = t_0;
	} else if (z <= -1.8e-243) {
		tmp = 4.0 * (x / y);
	} else if (z <= 6.5e+91) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * ((-4.0d0) / y)
    if (z <= (-1.2d+43)) then
        tmp = t_0
    else if (z <= (-1.8d-243)) then
        tmp = 4.0d0 * (x / y)
    else if (z <= 6.5d+91) then
        tmp = 4.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double tmp;
	if (z <= -1.2e+43) {
		tmp = t_0;
	} else if (z <= -1.8e-243) {
		tmp = 4.0 * (x / y);
	} else if (z <= 6.5e+91) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-4.0 / y)
	tmp = 0
	if z <= -1.2e+43:
		tmp = t_0
	elif z <= -1.8e-243:
		tmp = 4.0 * (x / y)
	elif z <= 6.5e+91:
		tmp = 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-4.0 / y))
	tmp = 0.0
	if (z <= -1.2e+43)
		tmp = t_0;
	elseif (z <= -1.8e-243)
		tmp = Float64(4.0 * Float64(x / y));
	elseif (z <= 6.5e+91)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-4.0 / y);
	tmp = 0.0;
	if (z <= -1.2e+43)
		tmp = t_0;
	elseif (z <= -1.8e-243)
		tmp = 4.0 * (x / y);
	elseif (z <= 6.5e+91)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.2e+43], t$95$0, If[LessEqual[z, -1.8e-243], N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.5e+91], 4.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.20000000000000012e43 or 6.4999999999999997e91 < z

   1. Initial program 99.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   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 y around 0 99.0%

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

      1. distribute-lft-out 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in z around inf 75.2%

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

      1. associate-*r/ 75.2%

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

      2. associate-*l/ 74.9%

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

      3. *-commutative 74.9%

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

   10. Simplified 74.9%

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

   if -1.20000000000000012e43 < z < -1.8000000000000001e-243

   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 66.1%

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

      1. *-commutative 66.1%

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

   7. Simplified 66.1%

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

   if -1.8000000000000001e-243 < z < 6.4999999999999997e91

   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 inf 52.4%

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

4. Final simplification 64.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+43}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-243}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+91}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := z \cdot \frac{-4}{y}\\ \mathbf{if}\;z \leq -4.1 \cdot 10^{+41}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -2.3 \cdot 10^{-243}:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+91}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* z (/ -4.0 y))))
   (if (<= z -4.1e+41)
     t_0
     (if (<= z -2.3e-243) (* x (/ 4.0 y)) (if (<= z 6.5e+91) 4.0 t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double tmp;
	if (z <= -4.1e+41) {
		tmp = t_0;
	} else if (z <= -2.3e-243) {
		tmp = x * (4.0 / y);
	} else if (z <= 6.5e+91) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = z * ((-4.0d0) / y)
    if (z <= (-4.1d+41)) then
        tmp = t_0
    else if (z <= (-2.3d-243)) then
        tmp = x * (4.0d0 / y)
    else if (z <= 6.5d+91) then
        tmp = 4.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = z * (-4.0 / y);
	double tmp;
	if (z <= -4.1e+41) {
		tmp = t_0;
	} else if (z <= -2.3e-243) {
		tmp = x * (4.0 / y);
	} else if (z <= 6.5e+91) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = z * (-4.0 / y)
	tmp = 0
	if z <= -4.1e+41:
		tmp = t_0
	elif z <= -2.3e-243:
		tmp = x * (4.0 / y)
	elif z <= 6.5e+91:
		tmp = 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(z * Float64(-4.0 / y))
	tmp = 0.0
	if (z <= -4.1e+41)
		tmp = t_0;
	elseif (z <= -2.3e-243)
		tmp = Float64(x * Float64(4.0 / y));
	elseif (z <= 6.5e+91)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = z * (-4.0 / y);
	tmp = 0.0;
	if (z <= -4.1e+41)
		tmp = t_0;
	elseif (z <= -2.3e-243)
		tmp = x * (4.0 / y);
	elseif (z <= 6.5e+91)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.1e+41], t$95$0, If[LessEqual[z, -2.3e-243], N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.5e+91], 4.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.1000000000000004e41 or 6.4999999999999997e91 < z

   1. Initial program 99.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   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 y around 0 99.0%

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

      1. distribute-lft-out 99.0%

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

   7. Simplified 99.0%

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

   8. Taylor expanded in z around inf 75.2%

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

      1. associate-*r/ 75.2%

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

      2. associate-*l/ 74.9%

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

      3. *-commutative 74.9%

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

   10. Simplified 74.9%

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

   if -4.1000000000000004e41 < z < -2.3e-243

   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 66.1%

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

      1. *-commutative 66.1%

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

      2. associate-*l/ 66.1%

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

      3. associate-*r/ 65.9%

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

   7. Simplified 65.9%

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

   if -2.3e-243 < z < 6.4999999999999997e91

   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 inf 52.4%

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

Alternative 5: 85.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+118} \lor \neg \left(y \leq 1.3 \cdot 10^{+69}\right):\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.7e+118) (not (<= y 1.3e+69)))
   (+ 4.0 (/ (* z -4.0) y))
   (* 4.0 (/ (- x z) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.7e+118) || !(y <= 1.3e+69)) {
		tmp = 4.0 + ((z * -4.0) / 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 <= (-1.7d+118)) .or. (.not. (y <= 1.3d+69))) then
        tmp = 4.0d0 + ((z * (-4.0d0)) / 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 <= -1.7e+118) || !(y <= 1.3e+69)) {
		tmp = 4.0 + ((z * -4.0) / y);
	} else {
		tmp = 4.0 * ((x - z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.7e+118) or not (y <= 1.3e+69):
		tmp = 4.0 + ((z * -4.0) / y)
	else:
		tmp = 4.0 * ((x - z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.7e+118) || !(y <= 1.3e+69))
		tmp = Float64(4.0 + Float64(Float64(z * -4.0) / 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 <= -1.7e+118) || ~((y <= 1.3e+69)))
		tmp = 4.0 + ((z * -4.0) / y);
	else
		tmp = 4.0 * ((x - z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.7e+118], N[Not[LessEqual[y, 1.3e+69]], $MachinePrecision]], N[(4.0 + N[(N[(z * -4.0), $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 < -1.69999999999999993e118 or 1.3000000000000001e69 < y

   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. +-commutative 98.6%

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

      2. associate-/l* 99.8%

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

      3. fma-define 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. remove-double-neg 99.8%

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

      7. sub-neg 99.8%

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

      8. associate--r+ 99.8%

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

      9. div-sub 99.8%

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

      10. sub-neg 99.8%

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

      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 x around 0 93.8%

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

      1. sub-neg 93.8%

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

      2. distribute-lft-in 93.8%

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

      3. metadata-eval 93.8%

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

      4. associate-+r+ 93.8%

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

      5. metadata-eval 93.8%

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

      6. neg-mul-1 93.8%

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

      7. associate-*r* 93.8%

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

      8. metadata-eval 93.8%

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

      9. *-commutative 93.8%

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

      10. associate-*l/ 93.8%

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

   7. Simplified 93.8%

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

   if -1.69999999999999993e118 < y < 1.3000000000000001e69

   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 y around 0 89.4%

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

      1. *-commutative 89.4%

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

   7. Simplified 89.4%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+118} \lor \neg \left(y \leq 1.3 \cdot 10^{+69}\right):\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+26} \lor \neg \left(z \leq 2.9 \cdot 10^{+92}\right):\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + x \cdot \frac{4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.05e+26) (not (<= z 2.9e+92)))
   (* 4.0 (/ (- x z) y))
   (+ 4.0 (* x (/ 4.0 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.05e+26) || !(z <= 2.9e+92)) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 4.0 + (x * (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 ((z <= (-1.05d+26)) .or. (.not. (z <= 2.9d+92))) then
        tmp = 4.0d0 * ((x - z) / y)
    else
        tmp = 4.0d0 + (x * (4.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.05e+26) || !(z <= 2.9e+92)) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 4.0 + (x * (4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.05e+26) or not (z <= 2.9e+92):
		tmp = 4.0 * ((x - z) / y)
	else:
		tmp = 4.0 + (x * (4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.05e+26) || !(z <= 2.9e+92))
		tmp = Float64(4.0 * Float64(Float64(x - z) / y));
	else
		tmp = Float64(4.0 + Float64(x * Float64(4.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.05e+26) || ~((z <= 2.9e+92)))
		tmp = 4.0 * ((x - z) / y);
	else
		tmp = 4.0 + (x * (4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.05e+26], N[Not[LessEqual[z, 2.9e+92]], $MachinePrecision]], N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05e26 or 2.9000000000000001e92 < z

   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. 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 y around 0 88.3%

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

      1. *-commutative 88.3%

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

   7. Simplified 88.3%

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

   if -1.05e26 < z < 2.9000000000000001e92

   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. +-commutative 99.9%

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

      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 88.7%

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

      1. distribute-lft-in 88.7%

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

      2. metadata-eval 88.7%

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

      3. associate-+r+ 88.7%

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

      4. metadata-eval 88.7%

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

      5. *-commutative 88.7%

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

      6. associate-*l/ 88.7%

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

      7. associate-*r/ 88.6%

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

   7. Simplified 88.6%

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

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+26} \lor \neg \left(z \leq 2.9 \cdot 10^{+92}\right):\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + x \cdot \frac{4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+132}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+125}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -7.8e+132) 4.0 (if (<= y 6e+125) (* 4.0 (/ (- x z) y)) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -7.8e+132) {
		tmp = 4.0;
	} else if (y <= 6e+125) {
		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 <= (-7.8d+132)) then
        tmp = 4.0d0
    else if (y <= 6d+125) 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 <= -7.8e+132) {
		tmp = 4.0;
	} else if (y <= 6e+125) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -7.8e+132:
		tmp = 4.0
	elif y <= 6e+125:
		tmp = 4.0 * ((x - z) / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -7.8e+132)
		tmp = 4.0;
	elseif (y <= 6e+125)
		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 <= -7.8e+132)
		tmp = 4.0;
	elseif (y <= 6e+125)
		tmp = 4.0 * ((x - z) / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -7.8e+132], 4.0, If[LessEqual[y, 6e+125], N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -7.80000000000000002e132 or 6.0000000000000003e125 < y

   1. Initial program 98.4%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
   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 79.5%

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

   if -7.80000000000000002e132 < y < 6.0000000000000003e125

   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 y around 0 87.4%

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

      1. *-commutative 87.4%

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

   7. Simplified 87.4%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{+132}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+125}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{+112}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+82}:\\ \;\;\;\;x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.5e+112) 4.0 (if (<= y 6.2e+82) (* x (/ 4.0 y)) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.5e+112) {
		tmp = 4.0;
	} else if (y <= 6.2e+82) {
		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 (y <= (-1.5d+112)) then
        tmp = 4.0d0
    else if (y <= 6.2d+82) 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 (y <= -1.5e+112) {
		tmp = 4.0;
	} else if (y <= 6.2e+82) {
		tmp = x * (4.0 / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.5e+112:
		tmp = 4.0
	elif y <= 6.2e+82:
		tmp = x * (4.0 / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.5e+112)
		tmp = 4.0;
	elseif (y <= 6.2e+82)
		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 (y <= -1.5e+112)
		tmp = 4.0;
	elseif (y <= 6.2e+82)
		tmp = x * (4.0 / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.5e+112], 4.0, If[LessEqual[y, 6.2e+82], N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4999999999999999e112 or 6.20000000000000065e82 < y

   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* 99.8%

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

      2. associate--l+ 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around inf 72.6%

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

   if -1.4999999999999999e112 < y < 6.20000000000000065e82

   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 inf 49.1%

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

      1. *-commutative 49.1%

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

      2. associate-*l/ 49.1%

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

      3. associate-*r/ 49.0%

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

   7. Simplified 49.0%

      \[\leadsto \color{blue}{x \cdot \frac{4}{y}} \]
3. Recombined 2 regimes into one program.
4. 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 99.6%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
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. Final simplification 99.9%

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

Alternative 10: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
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 0 99.6%

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

   1. distribute-lft-out 99.6%

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

7. Simplified 99.6%

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

8. Final simplification 99.6%

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

Alternative 11: 34.2% 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 99.6%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \]
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 29.9%

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

Reproduce

herbie shell --seed 2024145 
(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)))