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

Percentage Accurate: 99.9% → 100.0%

Time: 5.5s

Alternatives: 7

Speedup: 1.2×

• 21.6% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(-4.0 * N[(N[(N[(y - x), $MachinePrecision] / z), $MachinePrecision] - -0.5), $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. remove-double-neg 99.6%

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

   2. neg-mul-1 99.6%

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

   3. times-frac 100.0%

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

   4. metadata-eval 100.0%

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

   5. div-sub 100.0%

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

   6. distribute-frac-neg2 100.0%

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

   7. distribute-frac-neg 100.0%

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

   8. sub-neg 100.0%

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

   9. +-commutative 100.0%

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

   10. distribute-neg-out 100.0%

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

   11. remove-double-neg 100.0%

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

   12. sub-neg 100.0%

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

   13. *-commutative 100.0%

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

   14. neg-mul-1 100.0%

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

   15. times-frac 100.0%

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

   16. metadata-eval 100.0%

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

   17. *-inverses 100.0%

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

   18. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 54.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \frac{y}{z}\\ t_1 := \frac{x}{z} \cdot 4\\ \mathbf{if}\;y \leq -1.56 \cdot 10^{+119}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-74}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-162}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-83}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-19}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+57}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -4.0 (/ y z))) (t_1 (* (/ x z) 4.0)))
   (if (<= y -1.56e+119)
     t_0
     (if (<= y -6.8e-74)
       t_1
       (if (<= y 1.9e-162)
         -2.0
         (if (<= y 7.2e-83)
           t_1
           (if (<= y 1.45e-19)
             -2.0
             (if (<= y 1.36e+45) t_1 (if (<= y 5e+57) -2.0 t_0)))))))))

C

double code(double x, double y, double z) {
	double t_0 = -4.0 * (y / z);
	double t_1 = (x / z) * 4.0;
	double tmp;
	if (y <= -1.56e+119) {
		tmp = t_0;
	} else if (y <= -6.8e-74) {
		tmp = t_1;
	} else if (y <= 1.9e-162) {
		tmp = -2.0;
	} else if (y <= 7.2e-83) {
		tmp = t_1;
	} else if (y <= 1.45e-19) {
		tmp = -2.0;
	} else if (y <= 1.36e+45) {
		tmp = t_1;
	} else if (y <= 5e+57) {
		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) * (y / z)
    t_1 = (x / z) * 4.0d0
    if (y <= (-1.56d+119)) then
        tmp = t_0
    else if (y <= (-6.8d-74)) then
        tmp = t_1
    else if (y <= 1.9d-162) then
        tmp = -2.0d0
    else if (y <= 7.2d-83) then
        tmp = t_1
    else if (y <= 1.45d-19) then
        tmp = -2.0d0
    else if (y <= 1.36d+45) then
        tmp = t_1
    else if (y <= 5d+57) then
        tmp = -2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = -4.0 * (y / z);
	double t_1 = (x / z) * 4.0;
	double tmp;
	if (y <= -1.56e+119) {
		tmp = t_0;
	} else if (y <= -6.8e-74) {
		tmp = t_1;
	} else if (y <= 1.9e-162) {
		tmp = -2.0;
	} else if (y <= 7.2e-83) {
		tmp = t_1;
	} else if (y <= 1.45e-19) {
		tmp = -2.0;
	} else if (y <= 1.36e+45) {
		tmp = t_1;
	} else if (y <= 5e+57) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -4.0 * (y / z)
	t_1 = (x / z) * 4.0
	tmp = 0
	if y <= -1.56e+119:
		tmp = t_0
	elif y <= -6.8e-74:
		tmp = t_1
	elif y <= 1.9e-162:
		tmp = -2.0
	elif y <= 7.2e-83:
		tmp = t_1
	elif y <= 1.45e-19:
		tmp = -2.0
	elif y <= 1.36e+45:
		tmp = t_1
	elif y <= 5e+57:
		tmp = -2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-4.0 * Float64(y / z))
	t_1 = Float64(Float64(x / z) * 4.0)
	tmp = 0.0
	if (y <= -1.56e+119)
		tmp = t_0;
	elseif (y <= -6.8e-74)
		tmp = t_1;
	elseif (y <= 1.9e-162)
		tmp = -2.0;
	elseif (y <= 7.2e-83)
		tmp = t_1;
	elseif (y <= 1.45e-19)
		tmp = -2.0;
	elseif (y <= 1.36e+45)
		tmp = t_1;
	elseif (y <= 5e+57)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -4.0 * (y / z);
	t_1 = (x / z) * 4.0;
	tmp = 0.0;
	if (y <= -1.56e+119)
		tmp = t_0;
	elseif (y <= -6.8e-74)
		tmp = t_1;
	elseif (y <= 1.9e-162)
		tmp = -2.0;
	elseif (y <= 7.2e-83)
		tmp = t_1;
	elseif (y <= 1.45e-19)
		tmp = -2.0;
	elseif (y <= 1.36e+45)
		tmp = t_1;
	elseif (y <= 5e+57)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[y, -1.56e+119], t$95$0, If[LessEqual[y, -6.8e-74], t$95$1, If[LessEqual[y, 1.9e-162], -2.0, If[LessEqual[y, 7.2e-83], t$95$1, If[LessEqual[y, 1.45e-19], -2.0, If[LessEqual[y, 1.36e+45], t$95$1, If[LessEqual[y, 5e+57], -2.0, t$95$0]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.56e119 or 4.99999999999999972e57 < y

   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* 99.6%

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

      3. associate--l- 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 77.8%

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

      1. *-commutative 77.8%

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

   7. Simplified 77.8%

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

   if -1.56e119 < y < -6.8000000000000001e-74 or 1.90000000000000002e-162 < y < 7.20000000000000025e-83 or 1.45e-19 < y < 1.36e45

   1. Initial program 98.6%

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

      1. *-commutative 98.6%

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

      2. associate-/l* 99.7%

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

      3. associate--l- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 55.1%

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

   if -6.8000000000000001e-74 < y < 1.90000000000000002e-162 or 7.20000000000000025e-83 < y < 1.45e-19 or 1.36e45 < y < 4.99999999999999972e57

   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* 99.8%

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

      3. associate--l- 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 62.8%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.56 \cdot 10^{+119}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-74}:\\ \;\;\;\;\frac{x}{z} \cdot 4\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-162}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-83}:\\ \;\;\;\;\frac{x}{z} \cdot 4\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-19}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 1.36 \cdot 10^{+45}:\\ \;\;\;\;\frac{x}{z} \cdot 4\\ \mathbf{elif}\;y \leq 5 \cdot 10^{+57}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+119} \lor \neg \left(y \leq 2.65 \cdot 10^{+82}\right):\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(0.5 - \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.4e+119) (not (<= y 2.65e+82)))
   (* -4.0 (/ y z))
   (* -4.0 (- 0.5 (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.4e+119) || !(y <= 2.65e+82)) {
		tmp = -4.0 * (y / z);
	} else {
		tmp = -4.0 * (0.5 - (x / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-3.4d+119)) .or. (.not. (y <= 2.65d+82))) then
        tmp = (-4.0d0) * (y / z)
    else
        tmp = (-4.0d0) * (0.5d0 - (x / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.4e+119) || !(y <= 2.65e+82)) {
		tmp = -4.0 * (y / z);
	} else {
		tmp = -4.0 * (0.5 - (x / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.4e+119) or not (y <= 2.65e+82):
		tmp = -4.0 * (y / z)
	else:
		tmp = -4.0 * (0.5 - (x / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.4e+119) || !(y <= 2.65e+82))
		tmp = Float64(-4.0 * Float64(y / z));
	else
		tmp = Float64(-4.0 * Float64(0.5 - Float64(x / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.4e+119) || ~((y <= 2.65e+82)))
		tmp = -4.0 * (y / z);
	else
		tmp = -4.0 * (0.5 - (x / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.4e+119], N[Not[LessEqual[y, 2.65e+82]], $MachinePrecision]], N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(-4.0 * N[(0.5 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.40000000000000013e119 or 2.64999999999999989e82 < y

   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* 99.6%

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

      3. associate--l- 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 79.7%

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

      1. *-commutative 79.7%

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

   7. Simplified 79.7%

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

   if -3.40000000000000013e119 < y < 2.64999999999999989e82

   1. Initial program 99.4%

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

      1. remove-double-neg 99.4%

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

      2. neg-mul-1 99.4%

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

      3. times-frac 100.0%

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

      4. metadata-eval 100.0%

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

      5. div-sub 100.0%

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

      6. distribute-frac-neg2 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. +-commutative 100.0%

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

      10. distribute-neg-out 100.0%

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

      11. remove-double-neg 100.0%

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

      12. sub-neg 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. times-frac 100.0%

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

      16. metadata-eval 100.0%

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

      17. *-inverses 100.0%

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

      18. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 88.8%

      \[\leadsto -4 \cdot \left(\color{blue}{-1 \cdot \frac{x}{z}} - -0.5\right) \]
   6. Step-by-step derivation

      1. neg-mul-1 88.8%

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

      2. distribute-neg-frac 88.8%

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

   7. Simplified 88.8%

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

   8. Taylor expanded in x around 0 88.8%

      \[\leadsto -4 \cdot \color{blue}{\left(0.5 + -1 \cdot \frac{x}{z}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 88.8%

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

      2. unsub-neg 88.8%

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

   10. Simplified 88.8%

       \[\leadsto -4 \cdot \color{blue}{\left(0.5 - \frac{x}{z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.4 \cdot 10^{+119} \lor \neg \left(y \leq 2.65 \cdot 10^{+82}\right):\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(0.5 - \frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -95000000 \lor \neg \left(x \leq 2.7 \cdot 10^{+116}\right):\\ \;\;\;\;-4 \cdot \left(0.5 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -95000000.0) (not (<= x 2.7e+116)))
   (* -4.0 (- 0.5 (/ x z)))
   (* 4.0 (- -0.5 (/ y z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -95000000.0) || !(x <= 2.7e+116)) {
		tmp = -4.0 * (0.5 - (x / z));
	} else {
		tmp = 4.0 * (-0.5 - (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 ((x <= (-95000000.0d0)) .or. (.not. (x <= 2.7d+116))) then
        tmp = (-4.0d0) * (0.5d0 - (x / z))
    else
        tmp = 4.0d0 * ((-0.5d0) - (y / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -95000000.0) || !(x <= 2.7e+116)) {
		tmp = -4.0 * (0.5 - (x / z));
	} else {
		tmp = 4.0 * (-0.5 - (y / z));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -95000000.0) or not (x <= 2.7e+116):
		tmp = -4.0 * (0.5 - (x / z))
	else:
		tmp = 4.0 * (-0.5 - (y / z))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -95000000.0) || !(x <= 2.7e+116))
		tmp = Float64(-4.0 * Float64(0.5 - Float64(x / z)));
	else
		tmp = Float64(4.0 * Float64(-0.5 - Float64(y / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -95000000.0) || ~((x <= 2.7e+116)))
		tmp = -4.0 * (0.5 - (x / z));
	else
		tmp = 4.0 * (-0.5 - (y / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -95000000.0], N[Not[LessEqual[x, 2.7e+116]], $MachinePrecision]], N[(-4.0 * N[(0.5 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(-0.5 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.5e7 or 2.7e116 < x

   1. Initial program 99.0%

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

      1. remove-double-neg 99.0%

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

      2. neg-mul-1 99.0%

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

      3. times-frac 99.9%

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

      4. metadata-eval 99.9%

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

      5. div-sub 100.0%

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

      6. distribute-frac-neg2 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. +-commutative 100.0%

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

      10. distribute-neg-out 100.0%

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

      11. remove-double-neg 100.0%

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

      12. sub-neg 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. times-frac 100.0%

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

      16. metadata-eval 100.0%

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

      17. *-inverses 100.0%

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

      18. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 89.7%

      \[\leadsto -4 \cdot \left(\color{blue}{-1 \cdot \frac{x}{z}} - -0.5\right) \]
   6. Step-by-step derivation

      1. neg-mul-1 89.7%

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

      2. distribute-neg-frac 89.7%

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

   7. Simplified 89.7%

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

   8. Taylor expanded in x around 0 89.7%

      \[\leadsto -4 \cdot \color{blue}{\left(0.5 + -1 \cdot \frac{x}{z}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 89.7%

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

      2. unsub-neg 89.7%

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

   10. Simplified 89.7%

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

   if -9.5e7 < x < 2.7e116

   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* 99.7%

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

      3. associate--l- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 93.9%

      \[\leadsto \color{blue}{-4 \cdot \frac{y + 0.5 \cdot z}{z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 93.9%

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

      2. metadata-eval 93.9%

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

      3. +-commutative 93.9%

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

      4. *-commutative 93.9%

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

      5. fma-undefine 93.9%

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

      6. associate-*r* 93.9%

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

      7. neg-mul-1 93.9%

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

      8. associate-/l* 93.9%

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

      9. fma-undefine 93.9%

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

      10. *-commutative 93.9%

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

      11. distribute-neg-in 93.9%

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

      12. sub-neg 93.9%

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

      13. div-sub 93.9%

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

      14. distribute-neg-frac 93.9%

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

      15. associate-/l* 93.9%

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

      16. *-inverses 93.9%

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

      17. metadata-eval 93.9%

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

      18. metadata-eval 93.9%

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

   7. Simplified 93.9%

      \[\leadsto \color{blue}{4 \cdot \left(-0.5 - \frac{y}{z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -95000000 \lor \neg \left(x \leq 2.7 \cdot 10^{+116}\right):\\ \;\;\;\;-4 \cdot \left(0.5 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -10400000:\\ \;\;\;\;-4 \cdot \left(0.5 - \frac{x}{z}\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+84}:\\ \;\;\;\;4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -10400000.0)
   (* -4.0 (- 0.5 (/ x z)))
   (if (<= x 4.8e+84) (* 4.0 (- -0.5 (/ y z))) (* 4.0 (/ (- x y) z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -10400000.0) {
		tmp = -4.0 * (0.5 - (x / z));
	} else if (x <= 4.8e+84) {
		tmp = 4.0 * (-0.5 - (y / z));
	} 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 (x <= (-10400000.0d0)) then
        tmp = (-4.0d0) * (0.5d0 - (x / z))
    else if (x <= 4.8d+84) then
        tmp = 4.0d0 * ((-0.5d0) - (y / z))
    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 (x <= -10400000.0) {
		tmp = -4.0 * (0.5 - (x / z));
	} else if (x <= 4.8e+84) {
		tmp = 4.0 * (-0.5 - (y / z));
	} else {
		tmp = 4.0 * ((x - y) / z);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -10400000.0:
		tmp = -4.0 * (0.5 - (x / z))
	elif x <= 4.8e+84:
		tmp = 4.0 * (-0.5 - (y / z))
	else:
		tmp = 4.0 * ((x - y) / z)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -10400000.0)
		tmp = Float64(-4.0 * Float64(0.5 - Float64(x / z)));
	elseif (x <= 4.8e+84)
		tmp = Float64(4.0 * Float64(-0.5 - Float64(y / z)));
	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 (x <= -10400000.0)
		tmp = -4.0 * (0.5 - (x / z));
	elseif (x <= 4.8e+84)
		tmp = 4.0 * (-0.5 - (y / z));
	else
		tmp = 4.0 * ((x - y) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -10400000.0], N[(-4.0 * N[(0.5 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.8e+84], N[(4.0 * N[(-0.5 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.04e7

   1. Initial program 98.5%

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

      1. remove-double-neg 98.5%

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

      2. neg-mul-1 98.5%

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

      3. times-frac 99.9%

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

      4. metadata-eval 99.9%

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

      5. div-sub 100.0%

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

      6. distribute-frac-neg2 100.0%

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

      7. distribute-frac-neg 100.0%

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

      8. sub-neg 100.0%

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

      9. +-commutative 100.0%

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

      10. distribute-neg-out 100.0%

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

      11. remove-double-neg 100.0%

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

      12. sub-neg 100.0%

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

      13. *-commutative 100.0%

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

      14. neg-mul-1 100.0%

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

      15. times-frac 100.0%

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

      16. metadata-eval 100.0%

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

      17. *-inverses 100.0%

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

      18. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 87.7%

      \[\leadsto -4 \cdot \left(\color{blue}{-1 \cdot \frac{x}{z}} - -0.5\right) \]
   6. Step-by-step derivation

      1. neg-mul-1 87.7%

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

      2. distribute-neg-frac 87.7%

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

   7. Simplified 87.7%

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

   8. Taylor expanded in x around 0 87.7%

      \[\leadsto -4 \cdot \color{blue}{\left(0.5 + -1 \cdot \frac{x}{z}\right)} \]
   9. Step-by-step derivation

      1. mul-1-neg 87.7%

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

      2. unsub-neg 87.7%

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

   10. Simplified 87.7%

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

   if -1.04e7 < x < 4.7999999999999999e84

   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* 99.7%

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

      3. associate--l- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around 0 95.0%

      \[\leadsto \color{blue}{-4 \cdot \frac{y + 0.5 \cdot z}{z}} \]
   6. Step-by-step derivation

      1. associate-*r/ 95.0%

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

      2. metadata-eval 95.0%

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

      3. +-commutative 95.0%

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

      4. *-commutative 95.0%

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

      5. fma-undefine 95.0%

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

      6. associate-*r* 95.0%

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

      7. neg-mul-1 95.0%

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

      8. associate-/l* 95.0%

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

      9. fma-undefine 95.0%

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

      10. *-commutative 95.0%

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

      11. distribute-neg-in 95.0%

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

      12. sub-neg 95.0%

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

      13. div-sub 95.0%

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

      14. distribute-neg-frac 95.0%

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

      15. associate-/l* 95.0%

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

      16. *-inverses 95.0%

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

      17. metadata-eval 95.0%

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

      18. metadata-eval 95.0%

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

   7. Simplified 95.0%

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

   if 4.7999999999999999e84 < 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* 99.7%

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

      3. associate--l- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around 0 91.2%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -10400000:\\ \;\;\;\;-4 \cdot \left(0.5 - \frac{x}{z}\right)\\ \mathbf{elif}\;x \leq 4.8 \cdot 10^{+84}:\\ \;\;\;\;4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 54.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+69} \lor \neg \left(x \leq 5.6 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{x}{z} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.5e+69) (not (<= x 5.6e+82))) (* (/ x z) 4.0) -2.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.5e+69) || !(x <= 5.6e+82)) {
		tmp = (x / z) * 4.0;
	} 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 ((x <= (-4.5d+69)) .or. (.not. (x <= 5.6d+82))) then
        tmp = (x / z) * 4.0d0
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.5e+69) || !(x <= 5.6e+82)) {
		tmp = (x / z) * 4.0;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.5e+69) or not (x <= 5.6e+82):
		tmp = (x / z) * 4.0
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.5e+69) || !(x <= 5.6e+82))
		tmp = Float64(Float64(x / z) * 4.0);
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4.5e+69) || ~((x <= 5.6e+82)))
		tmp = (x / z) * 4.0;
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.5e+69], N[Not[LessEqual[x, 5.6e+82]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * 4.0), $MachinePrecision], -2.0]

Derivation

1. Split input into 2 regimes

2. if x < -4.4999999999999999e69 or 5.6000000000000001e82 < x

   1. Initial program 98.9%

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

      1. *-commutative 98.9%

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

      2. associate-/l* 99.7%

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

      3. associate--l- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 74.0%

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

   if -4.4999999999999999e69 < x < 5.6000000000000001e82

   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* 99.7%

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

      3. associate--l- 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in z around inf 46.0%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{+69} \lor \neg \left(x \leq 5.6 \cdot 10^{+82}\right):\\ \;\;\;\;\frac{x}{z} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.0% accurate, 11.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-/l* 99.7%

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

   3. associate--l- 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in z around inf 34.4%

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

6. Final simplification 34.4%

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

Developer target: 97.9% 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 2024055 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

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

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