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

Percentage Accurate: 99.8% → 99.8%

Time: 4.0s

Alternatives: 8

Speedup: 0.1×

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-/l* 100.0%

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

   3. div-sub 100.0%

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

   4. *-lft-identity 100.0%

      \[\leadsto \frac{x - y}{\frac{\color{blue}{1 \cdot z}}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

   5. metadata-eval 100.0%

      \[\leadsto \frac{x - y}{\frac{\color{blue}{\left(--1\right)} \cdot z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

   6. associate-/l* 99.9%

      \[\leadsto \frac{x - y}{\color{blue}{\frac{--1}{\frac{4}{z}}}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

   7. associate-/r/ 99.8%

      \[\leadsto \color{blue}{\frac{x - y}{--1} \cdot \frac{4}{z}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

   8. fma-neg 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - y}{--1}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right)} \]

   9. metadata-eval 99.8%

      \[\leadsto \mathsf{fma}\left(\frac{x - y}{\color{blue}{1}}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right) \]

   10. /-rgt-identity 99.8%

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

   11. associate-/r/ 99.8%

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

   12. distribute-lft-neg-in 99.8%

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

   13. distribute-frac-neg 99.8%

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

   14. *-commutative 99.8%

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

   15. distribute-lft-neg-in 99.8%

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

   16. associate-/l* 99.8%

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

   17. metadata-eval 99.8%

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

   18. *-inverses 99.8%

       \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{-0.5}{\color{blue}{1}} \cdot 4\right) \]

   19. metadata-eval 99.8%

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

   20. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, -2\right) \]
6. Add Preprocessing

Alternative 2: 53.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{z}\\ t_1 := -4 \cdot \frac{y}{z}\\ \mathbf{if}\;x \leq -1.05 \cdot 10^{+68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-226}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 2400000:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x z))) (t_1 (* -4.0 (/ y z))))
   (if (<= x -1.05e+68)
     t_0
     (if (<= x -2.1e-115)
       t_1
       (if (<= x 3e-226)
         -2.0
         (if (<= x 2.1e-98) t_1 (if (<= x 2400000.0) -2.0 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / z);
	double t_1 = -4.0 * (y / z);
	double tmp;
	if (x <= -1.05e+68) {
		tmp = t_0;
	} else if (x <= -2.1e-115) {
		tmp = t_1;
	} else if (x <= 3e-226) {
		tmp = -2.0;
	} else if (x <= 2.1e-98) {
		tmp = t_1;
	} else if (x <= 2400000.0) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = 4.0d0 * (x / z)
    t_1 = (-4.0d0) * (y / z)
    if (x <= (-1.05d+68)) then
        tmp = t_0
    else if (x <= (-2.1d-115)) then
        tmp = t_1
    else if (x <= 3d-226) then
        tmp = -2.0d0
    else if (x <= 2.1d-98) then
        tmp = t_1
    else if (x <= 2400000.0d0) then
        tmp = -2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / z);
	double t_1 = -4.0 * (y / z);
	double tmp;
	if (x <= -1.05e+68) {
		tmp = t_0;
	} else if (x <= -2.1e-115) {
		tmp = t_1;
	} else if (x <= 3e-226) {
		tmp = -2.0;
	} else if (x <= 2.1e-98) {
		tmp = t_1;
	} else if (x <= 2400000.0) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / z)
	t_1 = -4.0 * (y / z)
	tmp = 0
	if x <= -1.05e+68:
		tmp = t_0
	elif x <= -2.1e-115:
		tmp = t_1
	elif x <= 3e-226:
		tmp = -2.0
	elif x <= 2.1e-98:
		tmp = t_1
	elif x <= 2400000.0:
		tmp = -2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / z))
	t_1 = Float64(-4.0 * Float64(y / z))
	tmp = 0.0
	if (x <= -1.05e+68)
		tmp = t_0;
	elseif (x <= -2.1e-115)
		tmp = t_1;
	elseif (x <= 3e-226)
		tmp = -2.0;
	elseif (x <= 2.1e-98)
		tmp = t_1;
	elseif (x <= 2400000.0)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / z);
	t_1 = -4.0 * (y / z);
	tmp = 0.0;
	if (x <= -1.05e+68)
		tmp = t_0;
	elseif (x <= -2.1e-115)
		tmp = t_1;
	elseif (x <= 3e-226)
		tmp = -2.0;
	elseif (x <= 2.1e-98)
		tmp = t_1;
	elseif (x <= 2400000.0)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.05e+68], t$95$0, If[LessEqual[x, -2.1e-115], t$95$1, If[LessEqual[x, 3e-226], -2.0, If[LessEqual[x, 2.1e-98], t$95$1, If[LessEqual[x, 2400000.0], -2.0, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e68 or 2.4e6 < x

   1. Initial program 100.0%

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

      1. associate-*l/ 99.7%

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

      2. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.7%

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

      6. sub-neg 99.7%

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

      7. distribute-rgt-neg-in 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 69.4%

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

   if -1.05e68 < x < -2.10000000000000002e-115 or 2.99999999999999995e-226 < x < 2.09999999999999992e-98

   1. Initial program 98.7%

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

      1. associate-*l/ 99.7%

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

      2. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.7%

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

      6. sub-neg 99.7%

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

      7. distribute-rgt-neg-in 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 58.4%

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

   if -2.10000000000000002e-115 < x < 2.99999999999999995e-226 or 2.09999999999999992e-98 < x < 2.4e6

   1. Initial program 100.0%

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

      1. associate-*l/ 99.7%

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

      2. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.7%

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

      6. sub-neg 99.7%

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

      7. distribute-rgt-neg-in 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 59.1%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{+68}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{elif}\;x \leq -2.1 \cdot 10^{-115}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{-226}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-98}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{elif}\;x \leq 2400000:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -520000000000 \lor \neg \left(z \leq 1.45 \cdot 10^{+15}\right):\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -520000000000.0) (not (<= z 1.45e+15)))
   (- (* -4.0 (/ y z)) 2.0)
   (* 4.0 (/ (- x y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -520000000000.0) || !(z <= 1.45e+15)) {
		tmp = (-4.0 * (y / z)) - 2.0;
	} else {
		tmp = 4.0 * ((x - y) / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-520000000000.0d0)) .or. (.not. (z <= 1.45d+15))) then
        tmp = ((-4.0d0) * (y / z)) - 2.0d0
    else
        tmp = 4.0d0 * ((x - y) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -520000000000.0) || !(z <= 1.45e+15)) {
		tmp = (-4.0 * (y / z)) - 2.0;
	} else {
		tmp = 4.0 * ((x - y) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -520000000000.0) or not (z <= 1.45e+15):
		tmp = (-4.0 * (y / z)) - 2.0
	else:
		tmp = 4.0 * ((x - y) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -520000000000.0) || !(z <= 1.45e+15))
		tmp = Float64(Float64(-4.0 * Float64(y / z)) - 2.0);
	else
		tmp = Float64(4.0 * Float64(Float64(x - y) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -520000000000.0) || ~((z <= 1.45e+15)))
		tmp = (-4.0 * (y / z)) - 2.0;
	else
		tmp = 4.0 * ((x - y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -520000000000.0], N[Not[LessEqual[z, 1.45e+15]], $MachinePrecision]], N[(N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.2e11 or 1.45e15 < z

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. div-sub 100.0%

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

      4. *-lft-identity 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{1 \cdot z}}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{\left(--1\right)} \cdot z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      6. associate-/l* 99.9%

         \[\leadsto \frac{x - y}{\color{blue}{\frac{--1}{\frac{4}{z}}}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      7. associate-/r/ 99.8%

         \[\leadsto \color{blue}{\frac{x - y}{--1} \cdot \frac{4}{z}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      8. fma-neg 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - y}{--1}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right)} \]

      9. metadata-eval 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{x - y}{\color{blue}{1}}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right) \]

      10. /-rgt-identity 99.8%

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

      11. associate-/r/ 99.8%

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

      12. distribute-lft-neg-in 99.8%

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

      13. distribute-frac-neg 99.8%

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

      14. *-commutative 99.8%

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

      15. distribute-lft-neg-in 99.8%

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

      16. associate-/l* 99.8%

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

      17. metadata-eval 99.8%

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

      18. *-inverses 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{-0.5}{\color{blue}{1}} \cdot 4\right) \]

      19. metadata-eval 99.8%

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

      20. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 82.3%

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

   if -5.2e11 < z < 1.45e15

   1. Initial program 99.3%

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

      1. associate-*l/ 99.7%

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

      2. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.7%

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

      6. sub-neg 99.7%

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

      7. distribute-rgt-neg-in 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 95.1%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -520000000000 \lor \neg \left(z \leq 1.45 \cdot 10^{+15}\right):\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+187}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+143}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.05e+187) -2.0 (if (<= z 6.5e+143) (* 4.0 (/ (- x y) z)) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.05e+187) {
		tmp = -2.0;
	} else if (z <= 6.5e+143) {
		tmp = 4.0 * ((x - y) / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= (-2.05d+187)) then
        tmp = -2.0d0
    else if (z <= 6.5d+143) then
        tmp = 4.0d0 * ((x - y) / z)
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.05e+187) {
		tmp = -2.0;
	} else if (z <= 6.5e+143) {
		tmp = 4.0 * ((x - y) / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.05e+187:
		tmp = -2.0
	elif z <= 6.5e+143:
		tmp = 4.0 * ((x - y) / z)
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.05e+187)
		tmp = -2.0;
	elseif (z <= 6.5e+143)
		tmp = Float64(4.0 * Float64(Float64(x - y) / z));
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.05e+187)
		tmp = -2.0;
	elseif (z <= 6.5e+143)
		tmp = 4.0 * ((x - y) / z);
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.05e+187], -2.0, If[LessEqual[z, 6.5e+143], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if z < -2.05e187 or 6.4999999999999997e143 < z

   1. Initial program 99.9%

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

      1. associate-*l/ 99.8%

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

      2. /-rgt-identity 99.8%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.8%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.8%

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

      6. sub-neg 99.8%

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

      7. distribute-rgt-neg-in 99.8%

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

      8. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 82.3%

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

   if -2.05e187 < z < 6.4999999999999997e143

   1. Initial program 99.6%

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

      1. associate-*l/ 99.7%

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

      2. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.7%

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

      6. sub-neg 99.7%

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

      7. distribute-rgt-neg-in 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 82.7%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.05 \cdot 10^{+187}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+143}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+44}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-31}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{z} - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -3e+44)
   (* 4.0 (/ (- x y) z))
   (if (<= x 8e-31) (- (* -4.0 (/ y z)) 2.0) (- (* 4.0 (/ x z)) 2.0))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e+44) {
		tmp = 4.0 * ((x - y) / z);
	} else if (x <= 8e-31) {
		tmp = (-4.0 * (y / z)) - 2.0;
	} else {
		tmp = (4.0 * (x / z)) - 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (x <= (-3d+44)) then
        tmp = 4.0d0 * ((x - y) / z)
    else if (x <= 8d-31) then
        tmp = ((-4.0d0) * (y / z)) - 2.0d0
    else
        tmp = (4.0d0 * (x / z)) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -3e+44) {
		tmp = 4.0 * ((x - y) / z);
	} else if (x <= 8e-31) {
		tmp = (-4.0 * (y / z)) - 2.0;
	} else {
		tmp = (4.0 * (x / z)) - 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -3e+44:
		tmp = 4.0 * ((x - y) / z)
	elif x <= 8e-31:
		tmp = (-4.0 * (y / z)) - 2.0
	else:
		tmp = (4.0 * (x / z)) - 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -3e+44)
		tmp = Float64(4.0 * Float64(Float64(x - y) / z));
	elseif (x <= 8e-31)
		tmp = Float64(Float64(-4.0 * Float64(y / z)) - 2.0);
	else
		tmp = Float64(Float64(4.0 * Float64(x / z)) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -3e+44)
		tmp = 4.0 * ((x - y) / z);
	elseif (x <= 8e-31)
		tmp = (-4.0 * (y / z)) - 2.0;
	else
		tmp = (4.0 * (x / z)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -3e+44], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8e-31], N[(N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision], N[(N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.99999999999999987e44

   1. Initial program 100.0%

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

      1. associate-*l/ 99.7%

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

      2. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.7%

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

      6. sub-neg 99.7%

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

      7. distribute-rgt-neg-in 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 90.3%

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

   if -2.99999999999999987e44 < x < 8.000000000000001e-31

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-/l* 100.0%

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

      3. div-sub 100.0%

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

      4. *-lft-identity 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{1 \cdot z}}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{\left(--1\right)} \cdot z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      6. associate-/l* 99.9%

         \[\leadsto \frac{x - y}{\color{blue}{\frac{--1}{\frac{4}{z}}}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      7. associate-/r/ 99.8%

         \[\leadsto \color{blue}{\frac{x - y}{--1} \cdot \frac{4}{z}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      8. fma-neg 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - y}{--1}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right)} \]

      9. metadata-eval 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{x - y}{\color{blue}{1}}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right) \]

      10. /-rgt-identity 99.8%

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

      11. associate-/r/ 99.8%

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

      12. distribute-lft-neg-in 99.8%

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

      13. distribute-frac-neg 99.8%

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

      14. *-commutative 99.8%

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

      15. distribute-lft-neg-in 99.8%

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

      16. associate-/l* 99.8%

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

      17. metadata-eval 99.8%

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

      18. *-inverses 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{-0.5}{\color{blue}{1}} \cdot 4\right) \]

      19. metadata-eval 99.8%

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

      20. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 90.1%

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

   if 8.000000000000001e-31 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/l* 100.0%

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

      3. div-sub 100.0%

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

      4. *-lft-identity 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{1 \cdot z}}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      5. metadata-eval 100.0%

         \[\leadsto \frac{x - y}{\frac{\color{blue}{\left(--1\right)} \cdot z}{4}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      6. associate-/l* 99.8%

         \[\leadsto \frac{x - y}{\color{blue}{\frac{--1}{\frac{4}{z}}}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      7. associate-/r/ 99.8%

         \[\leadsto \color{blue}{\frac{x - y}{--1} \cdot \frac{4}{z}} - \frac{z \cdot 0.5}{\frac{z}{4}} \]

      8. fma-neg 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - y}{--1}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right)} \]

      9. metadata-eval 99.8%

         \[\leadsto \mathsf{fma}\left(\frac{x - y}{\color{blue}{1}}, \frac{4}{z}, -\frac{z \cdot 0.5}{\frac{z}{4}}\right) \]

      10. /-rgt-identity 99.8%

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

      11. associate-/r/ 99.8%

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

      12. distribute-lft-neg-in 99.8%

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

      13. distribute-frac-neg 99.8%

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

      14. *-commutative 99.8%

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

      15. distribute-lft-neg-in 99.8%

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

      16. associate-/l* 99.8%

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

      17. metadata-eval 99.8%

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

      18. *-inverses 99.8%

          \[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, \frac{-0.5}{\color{blue}{1}} \cdot 4\right) \]

      19. metadata-eval 99.8%

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

      20. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 91.1%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+44}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \mathbf{elif}\;x \leq 8 \cdot 10^{-31}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{z} - 2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-37} \lor \neg \left(y \leq 1.35 \cdot 10^{+94}\right):\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.25e-37) (not (<= y 1.35e+94))) (* -4.0 (/ y z)) -2.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.25e-37) || !(y <= 1.35e+94)) {
		tmp = -4.0 * (y / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-1.25d-37)) .or. (.not. (y <= 1.35d+94))) then
        tmp = (-4.0d0) * (y / z)
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.25e-37) || !(y <= 1.35e+94)) {
		tmp = -4.0 * (y / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.25e-37) or not (y <= 1.35e+94):
		tmp = -4.0 * (y / z)
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.25e-37) || !(y <= 1.35e+94))
		tmp = Float64(-4.0 * Float64(y / z));
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.25e-37) || ~((y <= 1.35e+94)))
		tmp = -4.0 * (y / z);
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.25e-37], N[Not[LessEqual[y, 1.35e+94]], $MachinePrecision]], N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision], -2.0]

Derivation

1. Split input into 2 regimes

2. if y < -1.2499999999999999e-37 or 1.3500000000000001e94 < y

   1. Initial program 99.3%

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

      1. associate-*l/ 99.7%

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

      2. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.7%

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

      6. sub-neg 99.7%

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

      7. distribute-rgt-neg-in 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 61.5%

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

   if -1.2499999999999999e-37 < y < 1.3500000000000001e94

   1. Initial program 100.0%

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

      1. associate-*l/ 99.7%

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

      2. /-rgt-identity 99.7%

         \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

      4. metadata-eval 99.7%

         \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

      5. /-rgt-identity 99.7%

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

      6. sub-neg 99.7%

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

      7. distribute-rgt-neg-in 99.7%

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

      8. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 46.1%

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

4. Final simplification 53.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-37} \lor \neg \left(y \leq 1.35 \cdot 10^{+94}\right):\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / z) * ((x - y) + (z * (-0.5d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.7%

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

   2. /-rgt-identity 99.7%

      \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

   3. metadata-eval 99.7%

      \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

   4. metadata-eval 99.7%

      \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

   5. /-rgt-identity 99.7%

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

   6. sub-neg 99.7%

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

   7. distribute-rgt-neg-in 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Final simplification 99.7%

   \[\leadsto \frac{4}{z} \cdot \left(\left(x - y\right) + z \cdot -0.5\right) \]
6. Add Preprocessing

Alternative 8: 33.9% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-*l/ 99.7%

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

   2. /-rgt-identity 99.7%

      \[\leadsto \frac{4}{z} \cdot \color{blue}{\frac{\left(x - y\right) - z \cdot 0.5}{1}} \]

   3. metadata-eval 99.7%

      \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{--1}} \]

   4. metadata-eval 99.7%

      \[\leadsto \frac{4}{z} \cdot \frac{\left(x - y\right) - z \cdot 0.5}{\color{blue}{1}} \]

   5. /-rgt-identity 99.7%

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

   6. sub-neg 99.7%

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

   7. distribute-rgt-neg-in 99.7%

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

   8. metadata-eval 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in z around inf 32.1%

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

6. Final simplification 32.1%

   \[\leadsto -2 \]
7. Add Preprocessing

Developer target: 97.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * (x / z)) - (2.0d0 + (4.0d0 * (y / z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024019 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

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

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