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

Percentage Accurate: 99.8% → 100.0%

Time: 3.9s

Alternatives: 6

Speedup: 1.2×

• 20.6% 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: 99.8% 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: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{x - y}{z} - 2 \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) / z)) - 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 = (4.0d0 * ((x - y) / z)) - 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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* 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in z around 0 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 53.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{z}\\ t_1 := \frac{y \cdot -4}{z}\\ \mathbf{if}\;x \leq -11.8:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-252}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-245}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-233}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-11}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x z))) (t_1 (/ (* y -4.0) z)))
   (if (<= x -11.8)
     t_0
     (if (<= x -8.2e-252)
       t_1
       (if (<= x 1.7e-245)
         -2.0
         (if (<= x 6e-233) t_1 (if (<= x 9e-11) -2.0 t_0)))))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / z);
	double t_1 = (y * -4.0) / z;
	double tmp;
	if (x <= -11.8) {
		tmp = t_0;
	} else if (x <= -8.2e-252) {
		tmp = t_1;
	} else if (x <= 1.7e-245) {
		tmp = -2.0;
	} else if (x <= 6e-233) {
		tmp = t_1;
	} else if (x <= 9e-11) {
		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 = (y * (-4.0d0)) / z
    if (x <= (-11.8d0)) then
        tmp = t_0
    else if (x <= (-8.2d-252)) then
        tmp = t_1
    else if (x <= 1.7d-245) then
        tmp = -2.0d0
    else if (x <= 6d-233) then
        tmp = t_1
    else if (x <= 9d-11) 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 = (y * -4.0) / z;
	double tmp;
	if (x <= -11.8) {
		tmp = t_0;
	} else if (x <= -8.2e-252) {
		tmp = t_1;
	} else if (x <= 1.7e-245) {
		tmp = -2.0;
	} else if (x <= 6e-233) {
		tmp = t_1;
	} else if (x <= 9e-11) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / z)
	t_1 = (y * -4.0) / z
	tmp = 0
	if x <= -11.8:
		tmp = t_0
	elif x <= -8.2e-252:
		tmp = t_1
	elif x <= 1.7e-245:
		tmp = -2.0
	elif x <= 6e-233:
		tmp = t_1
	elif x <= 9e-11:
		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(Float64(y * -4.0) / z)
	tmp = 0.0
	if (x <= -11.8)
		tmp = t_0;
	elseif (x <= -8.2e-252)
		tmp = t_1;
	elseif (x <= 1.7e-245)
		tmp = -2.0;
	elseif (x <= 6e-233)
		tmp = t_1;
	elseif (x <= 9e-11)
		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 = (y * -4.0) / z;
	tmp = 0.0;
	if (x <= -11.8)
		tmp = t_0;
	elseif (x <= -8.2e-252)
		tmp = t_1;
	elseif (x <= 1.7e-245)
		tmp = -2.0;
	elseif (x <= 6e-233)
		tmp = t_1;
	elseif (x <= 9e-11)
		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[(N[(y * -4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[x, -11.8], t$95$0, If[LessEqual[x, -8.2e-252], t$95$1, If[LessEqual[x, 1.7e-245], -2.0, If[LessEqual[x, 6e-233], t$95$1, If[LessEqual[x, 9e-11], -2.0, t$95$0]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -11.800000000000001 or 8.9999999999999999e-11 < 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* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in x around inf 71.2%

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

   if -11.800000000000001 < x < -8.20000000000000028e-252 or 1.7e-245 < x < 5.99999999999999997e-233

   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* 99.6%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in y around inf 64.3%

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

      1. associate-*r/ 64.3%

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

   6. Simplified 64.3%

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

   if -8.20000000000000028e-252 < x < 1.7e-245 or 5.99999999999999997e-233 < x < 8.9999999999999999e-11

   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* 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around inf 62.4%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -11.8:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-252}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-245}:\\ \;\;\;\;-2\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-233}:\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-11}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \end{array} \]

Alternative 3: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+103} \lor \neg \left(y \leq 2.45 \cdot 10^{+135}\right):\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{z} + -2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.05e+103) (not (<= y 2.45e+135)))
   (/ (* y -4.0) z)
   (+ (* 4.0 (/ x z)) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.05e+103) || !(y <= 2.45e+135)) {
		tmp = (y * -4.0) / z;
	} 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 ((y <= (-1.05d+103)) .or. (.not. (y <= 2.45d+135))) then
        tmp = (y * (-4.0d0)) / z
    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 ((y <= -1.05e+103) || !(y <= 2.45e+135)) {
		tmp = (y * -4.0) / z;
	} else {
		tmp = (4.0 * (x / z)) + -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.05e+103) or not (y <= 2.45e+135):
		tmp = (y * -4.0) / z
	else:
		tmp = (4.0 * (x / z)) + -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.05e+103) || !(y <= 2.45e+135))
		tmp = Float64(Float64(y * -4.0) / z);
	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 ((y <= -1.05e+103) || ~((y <= 2.45e+135)))
		tmp = (y * -4.0) / z;
	else
		tmp = (4.0 * (x / z)) + -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.05e+103], N[Not[LessEqual[y, 2.45e+135]], $MachinePrecision]], N[(N[(y * -4.0), $MachinePrecision] / z), $MachinePrecision], N[(N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.0500000000000001e103 or 2.4500000000000001e135 < y

   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* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around inf 77.1%

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

      1. associate-*r/ 77.1%

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

   6. Simplified 77.1%

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

   if -1.0500000000000001e103 < y < 2.4500000000000001e135

   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* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 87.2%

      \[\leadsto \color{blue}{4 \cdot \frac{x - 0.5 \cdot z}{z}} \]
   5. Step-by-step derivation

      1. div-sub 87.2%

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

      2. sub-neg 87.2%

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

      3. associate-/l* 87.2%

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

      4. *-inverses 87.2%

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

      5. metadata-eval 87.2%

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

      6. metadata-eval 87.2%

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

      7. distribute-lft-in 87.2%

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

      8. metadata-eval 87.2%

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

   6. Simplified 87.2%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+103} \lor \neg \left(y \leq 2.45 \cdot 10^{+135}\right):\\ \;\;\;\;\frac{y \cdot -4}{z}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{z} + -2\\ \end{array} \]

Alternative 4: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -45 \lor \neg \left(x \leq 1.5 \cdot 10^{-5}\right):\\ \;\;\;\;4 \cdot \frac{x}{z} + -2\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -45.0) (not (<= x 1.5e-5)))
   (+ (* 4.0 (/ x z)) -2.0)
   (- (* -4.0 (/ y z)) 2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -45.0) || !(x <= 1.5e-5)) {
		tmp = (4.0 * (x / z)) + -2.0;
	} else {
		tmp = (-4.0 * (y / 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 <= (-45.0d0)) .or. (.not. (x <= 1.5d-5))) then
        tmp = (4.0d0 * (x / z)) + (-2.0d0)
    else
        tmp = ((-4.0d0) * (y / z)) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -45.0) || !(x <= 1.5e-5)) {
		tmp = (4.0 * (x / z)) + -2.0;
	} else {
		tmp = (-4.0 * (y / z)) - 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -45.0) or not (x <= 1.5e-5):
		tmp = (4.0 * (x / z)) + -2.0
	else:
		tmp = (-4.0 * (y / z)) - 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -45.0) || !(x <= 1.5e-5))
		tmp = Float64(Float64(4.0 * Float64(x / z)) + -2.0);
	else
		tmp = Float64(Float64(-4.0 * Float64(y / z)) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -45.0) || ~((x <= 1.5e-5)))
		tmp = (4.0 * (x / z)) + -2.0;
	else
		tmp = (-4.0 * (y / z)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -45.0], N[Not[LessEqual[x, 1.5e-5]], $MachinePrecision]], N[(N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision] + -2.0), $MachinePrecision], N[(N[(-4.0 * N[(y / z), $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -45 or 1.50000000000000004e-5 < 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* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 88.9%

      \[\leadsto \color{blue}{4 \cdot \frac{x - 0.5 \cdot z}{z}} \]
   5. Step-by-step derivation

      1. div-sub 88.9%

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

      2. sub-neg 88.9%

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

      3. associate-/l* 88.9%

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

      4. *-inverses 88.9%

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

      5. metadata-eval 88.9%

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

      6. metadata-eval 88.9%

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

      7. distribute-lft-in 88.9%

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

      8. metadata-eval 88.9%

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

   6. Simplified 88.9%

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

   if -45 < x < 1.50000000000000004e-5

   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* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 100.0%

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

   5. Taylor expanded in x around 0 94.6%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -45 \lor \neg \left(x \leq 1.5 \cdot 10^{-5}\right):\\ \;\;\;\;4 \cdot \frac{x}{z} + -2\\ \mathbf{else}:\\ \;\;\;\;-4 \cdot \frac{y}{z} - 2\\ \end{array} \]

Alternative 5: 53.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43} \lor \neg \left(x \leq 2.9 \cdot 10^{-11}\right):\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -5e-43) (not (<= x 2.9e-11))) (* 4.0 (/ x z)) -2.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -5e-43) || !(x <= 2.9e-11)) {
		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 ((x <= (-5d-43)) .or. (.not. (x <= 2.9d-11))) 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 ((x <= -5e-43) || !(x <= 2.9e-11)) {
		tmp = 4.0 * (x / z);
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -5e-43) or not (x <= 2.9e-11):
		tmp = 4.0 * (x / z)
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -5e-43) || !(x <= 2.9e-11))
		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 ((x <= -5e-43) || ~((x <= 2.9e-11)))
		tmp = 4.0 * (x / z);
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -5e-43], N[Not[LessEqual[x, 2.9e-11]], $MachinePrecision]], N[(4.0 * N[(x / z), $MachinePrecision]), $MachinePrecision], -2.0]

Derivation

1. Split input into 2 regimes

2. if x < -5.00000000000000019e-43 or 2.9e-11 < 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%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in x around inf 67.5%

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

   if -5.00000000000000019e-43 < x < 2.9e-11

   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* 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around inf 53.1%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-43} \lor \neg \left(x \leq 2.9 \cdot 10^{-11}\right):\\ \;\;\;\;4 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 6: 34.9% 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 100.0%

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

   1. associate-/l* 99.8%

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

3. Simplified 99.8%

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

4. Taylor expanded in z around inf 33.4%

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

5. Final simplification 33.4%

   \[\leadsto -2 \]

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 2023178 
(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))