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

Percentage Accurate: 99.9% → 100.0%

Time: 5.0s

Alternatives: 7

Speedup: 1.2×

• 20.5% 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. Add Preprocessing

Alternative 2: 53.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := -4 \cdot \frac{y}{z}\\ t_1 := 4 \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+77}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-110}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-212}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+78}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* -4.0 (/ y z))) (t_1 (* 4.0 (/ x z))))
   (if (<= z -1.2e+77)
     -2.0
     (if (<= z -1.1e-50)
       t_1
       (if (<= z -9e-110)
         t_0
         (if (<= z -5.8e-128)
           t_1
           (if (<= z 3.5e-212) t_0 (if (<= z 1.15e+78) t_1 -2.0))))))))

C

double code(double x, double y, double z) {
	double t_0 = -4.0 * (y / z);
	double t_1 = 4.0 * (x / z);
	double tmp;
	if (z <= -1.2e+77) {
		tmp = -2.0;
	} else if (z <= -1.1e-50) {
		tmp = t_1;
	} else if (z <= -9e-110) {
		tmp = t_0;
	} else if (z <= -5.8e-128) {
		tmp = t_1;
	} else if (z <= 3.5e-212) {
		tmp = t_0;
	} else if (z <= 1.15e+78) {
		tmp = t_1;
	} 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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (-4.0d0) * (y / z)
    t_1 = 4.0d0 * (x / z)
    if (z <= (-1.2d+77)) then
        tmp = -2.0d0
    else if (z <= (-1.1d-50)) then
        tmp = t_1
    else if (z <= (-9d-110)) then
        tmp = t_0
    else if (z <= (-5.8d-128)) then
        tmp = t_1
    else if (z <= 3.5d-212) then
        tmp = t_0
    else if (z <= 1.15d+78) then
        tmp = t_1
    else
        tmp = -2.0d0
    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 = 4.0 * (x / z);
	double tmp;
	if (z <= -1.2e+77) {
		tmp = -2.0;
	} else if (z <= -1.1e-50) {
		tmp = t_1;
	} else if (z <= -9e-110) {
		tmp = t_0;
	} else if (z <= -5.8e-128) {
		tmp = t_1;
	} else if (z <= 3.5e-212) {
		tmp = t_0;
	} else if (z <= 1.15e+78) {
		tmp = t_1;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = -4.0 * (y / z)
	t_1 = 4.0 * (x / z)
	tmp = 0
	if z <= -1.2e+77:
		tmp = -2.0
	elif z <= -1.1e-50:
		tmp = t_1
	elif z <= -9e-110:
		tmp = t_0
	elif z <= -5.8e-128:
		tmp = t_1
	elif z <= 3.5e-212:
		tmp = t_0
	elif z <= 1.15e+78:
		tmp = t_1
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(-4.0 * Float64(y / z))
	t_1 = Float64(4.0 * Float64(x / z))
	tmp = 0.0
	if (z <= -1.2e+77)
		tmp = -2.0;
	elseif (z <= -1.1e-50)
		tmp = t_1;
	elseif (z <= -9e-110)
		tmp = t_0;
	elseif (z <= -5.8e-128)
		tmp = t_1;
	elseif (z <= 3.5e-212)
		tmp = t_0;
	elseif (z <= 1.15e+78)
		tmp = t_1;
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = -4.0 * (y / z);
	t_1 = 4.0 * (x / z);
	tmp = 0.0;
	if (z <= -1.2e+77)
		tmp = -2.0;
	elseif (z <= -1.1e-50)
		tmp = t_1;
	elseif (z <= -9e-110)
		tmp = t_0;
	elseif (z <= -5.8e-128)
		tmp = t_1;
	elseif (z <= 3.5e-212)
		tmp = t_0;
	elseif (z <= 1.15e+78)
		tmp = t_1;
	else
		tmp = -2.0;
	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[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.2e+77], -2.0, If[LessEqual[z, -1.1e-50], t$95$1, If[LessEqual[z, -9e-110], t$95$0, If[LessEqual[z, -5.8e-128], t$95$1, If[LessEqual[z, 3.5e-212], t$95$0, If[LessEqual[z, 1.15e+78], t$95$1, -2.0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1999999999999999e77 or 1.1500000000000001e78 < z

   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* 99.6%

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

      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 z around inf 70.8%

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

   if -1.1999999999999999e77 < z < -1.0999999999999999e-50 or -9.0000000000000002e-110 < z < -5.8000000000000001e-128 or 3.4999999999999998e-212 < z < 1.1500000000000001e78

   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 54.4%

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

   if -1.0999999999999999e-50 < z < -9.0000000000000002e-110 or -5.8000000000000001e-128 < z < 3.4999999999999998e-212

   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* 99.8%

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

      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 y around inf 66.2%

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

      1. *-commutative 66.2%

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

   7. Simplified 66.2%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.2 \cdot 10^{+77}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-50}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq -9 \cdot 10^{-110}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq -5.8 \cdot 10^{-128}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-212}:\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+78}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+230} \lor \neg \left(y \leq -1.2 \cdot 10^{+208} \lor \neg \left(y \leq -1.25 \cdot 10^{+133}\right) \land y \leq 2.8 \cdot 10^{-8}\right):\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(0.5 - \frac{x}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.8e+230)
         (not
          (or (<= y -1.2e+208) (and (not (<= y -1.25e+133)) (<= y 2.8e-8)))))
   (* -4.0 (/ y z))
   (* -4.0 (- 0.5 (/ x z)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.8e+230) || !((y <= -1.2e+208) || (!(y <= -1.25e+133) && (y <= 2.8e-8)))) {
		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 <= (-3.8d+230)) .or. (.not. (y <= (-1.2d+208)) .or. (.not. (y <= (-1.25d+133))) .and. (y <= 2.8d-8))) 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 <= -3.8e+230) || !((y <= -1.2e+208) || (!(y <= -1.25e+133) && (y <= 2.8e-8)))) {
		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 <= -3.8e+230) or not ((y <= -1.2e+208) or (not (y <= -1.25e+133) and (y <= 2.8e-8))):
		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 <= -3.8e+230) || !((y <= -1.2e+208) || (!(y <= -1.25e+133) && (y <= 2.8e-8))))
		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 <= -3.8e+230) || ~(((y <= -1.2e+208) || (~((y <= -1.25e+133)) && (y <= 2.8e-8)))))
		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, -3.8e+230], N[Not[Or[LessEqual[y, -1.2e+208], And[N[Not[LessEqual[y, -1.25e+133]], $MachinePrecision], LessEqual[y, 2.8e-8]]]], $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 < -3.8e230 or -1.19999999999999993e208 < y < -1.2499999999999999e133 or 2.7999999999999999e-8 < y

   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. *-commutative 99.0%

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

      2. associate-/l* 99.5%

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

      3. associate--l- 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in y around inf 73.4%

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

      1. *-commutative 73.4%

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

   7. Simplified 73.4%

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

   if -3.8e230 < y < -1.19999999999999993e208 or -1.2499999999999999e133 < y < 2.7999999999999999e-8

   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. remove-double-neg 100.0%

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

      2. neg-mul-1 100.0%

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

      3. times-frac 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 88.0%

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

      1. neg-mul-1 88.0%

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

      2. distribute-neg-frac 88.0%

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

   7. Simplified 88.0%

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

   8. Taylor expanded in x around 0 88.0%

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

      1. mul-1-neg 88.0%

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

      2. unsub-neg 88.0%

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

   10. Simplified 88.0%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+230} \lor \neg \left(y \leq -1.2 \cdot 10^{+208} \lor \neg \left(y \leq -1.25 \cdot 10^{+133}\right) \land y \leq 2.8 \cdot 10^{-8}\right):\\ \;\;\;\;-4 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \left(0.5 - \frac{x}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+74} \lor \neg \left(z \leq 8.2 \cdot 10^{+80}\right):\\ \;\;\;\;4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1e+74) (not (<= z 8.2e+80)))
   (* 4.0 (- -0.5 (/ y z)))
   (* 4.0 (/ (- x y) z))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1e+74) || !(z <= 8.2e+80)) {
		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 ((z <= (-1d+74)) .or. (.not. (z <= 8.2d+80))) 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 ((z <= -1e+74) || !(z <= 8.2e+80)) {
		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 (z <= -1e+74) or not (z <= 8.2e+80):
		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 ((z <= -1e+74) || !(z <= 8.2e+80))
		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 ((z <= -1e+74) || ~((z <= 8.2e+80)))
		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[Or[LessEqual[z, -1e+74], N[Not[LessEqual[z, 8.2e+80]], $MachinePrecision]], 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 2 regimes

2. if z < -9.99999999999999952e73 or 8.20000000000000003e80 < z

   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* 99.6%

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

      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 x around 0 88.5%

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

      1. associate-*r/ 88.5%

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

      2. metadata-eval 88.5%

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

      3. +-commutative 88.5%

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

      4. *-commutative 88.5%

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

      5. fma-undefine 88.5%

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

      6. associate-*r* 88.5%

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

      7. neg-mul-1 88.5%

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

      8. associate-/l* 88.5%

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

      9. fma-undefine 88.5%

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

      10. *-commutative 88.5%

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

      11. distribute-neg-in 88.5%

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

      12. sub-neg 88.5%

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

      13. div-sub 88.5%

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

      14. distribute-neg-frac 88.5%

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

      15. associate-/l* 88.5%

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

      16. *-inverses 88.5%

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

      17. metadata-eval 88.5%

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

      18. metadata-eval 88.5%

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

   7. Simplified 88.5%

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

   if -9.99999999999999952e73 < z < 8.20000000000000003e80

   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. *-commutative 99.4%

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

      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 90.6%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+74} \lor \neg \left(z \leq 8.2 \cdot 10^{+80}\right):\\ \;\;\;\;4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+67} \lor \neg \left(x \leq 0.017\right):\\ \;\;\;\;-4 \cdot \left(0.5 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.4e+67) (not (<= x 0.017)))
   (* -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 <= -4.4e+67) || !(x <= 0.017)) {
		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 <= (-4.4d+67)) .or. (.not. (x <= 0.017d0))) 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 <= -4.4e+67) || !(x <= 0.017)) {
		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 <= -4.4e+67) or not (x <= 0.017):
		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 <= -4.4e+67) || !(x <= 0.017))
		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 <= -4.4e+67) || ~((x <= 0.017)))
		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, -4.4e+67], N[Not[LessEqual[x, 0.017]], $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 < -4.4e67 or 0.017000000000000001 < x

   1. Initial program 99.1%

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

      1. remove-double-neg 99.1%

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

      2. neg-mul-1 99.1%

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

      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 86.9%

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

      1. neg-mul-1 86.9%

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

      2. distribute-neg-frac 86.9%

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

   7. Simplified 86.9%

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

   8. Taylor expanded in x around 0 86.9%

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

      1. mul-1-neg 86.9%

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

      2. unsub-neg 86.9%

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

   10. Simplified 86.9%

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

   if -4.4e67 < x < 0.017000000000000001

   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* 99.6%

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

      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 x around 0 90.5%

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

      1. associate-*r/ 90.5%

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

      2. metadata-eval 90.5%

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

      3. +-commutative 90.5%

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

      4. *-commutative 90.5%

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

      5. fma-undefine 90.5%

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

      6. associate-*r* 90.5%

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

      7. neg-mul-1 90.5%

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

      8. associate-/l* 90.5%

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

      9. fma-undefine 90.5%

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

      10. *-commutative 90.5%

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

      11. distribute-neg-in 90.5%

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

      12. sub-neg 90.5%

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

      13. div-sub 90.5%

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

      14. distribute-neg-frac 90.5%

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

      15. associate-/l* 90.5%

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

      16. *-inverses 90.5%

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

      17. metadata-eval 90.5%

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

      18. metadata-eval 90.5%

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

   7. Simplified 90.5%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{+67} \lor \neg \left(x \leq 0.017\right):\\ \;\;\;\;-4 \cdot \left(0.5 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \left(-0.5 - \frac{y}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 53.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+71}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+80}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -1.35e+71) -2.0 (if (<= z 3.7e+80) (* 4.0 (/ x z)) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -1.35e+71) {
		tmp = -2.0;
	} else if (z <= 3.7e+80) {
		tmp = 4.0 * (x / 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 <= (-1.35d+71)) then
        tmp = -2.0d0
    else if (z <= 3.7d+80) then
        tmp = 4.0d0 * (x / 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 <= -1.35e+71) {
		tmp = -2.0;
	} else if (z <= 3.7e+80) {
		tmp = 4.0 * (x / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -1.35e+71:
		tmp = -2.0
	elif z <= 3.7e+80:
		tmp = 4.0 * (x / z)
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -1.35e+71)
		tmp = -2.0;
	elseif (z <= 3.7e+80)
		tmp = Float64(4.0 * Float64(x / z));
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -1.35e+71)
		tmp = -2.0;
	elseif (z <= 3.7e+80)
		tmp = 4.0 * (x / z);
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -1.35e+71], -2.0, If[LessEqual[z, 3.7e+80], N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if z < -1.34999999999999998e71 or 3.69999999999999996e80 < z

   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* 99.6%

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

      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 z around inf 70.8%

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

   if -1.34999999999999998e71 < z < 3.69999999999999996e80

   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. *-commutative 99.4%

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

      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 47.5%

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

Alternative 7: 34.8% 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 34.8%

   \[\leadsto \color{blue}{-2} \]
6. Add Preprocessing

Developer target: 97.7% 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 2024091 
(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))