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

Percentage Accurate: 99.8% → 99.8%

Time: 4.9s

Alternatives: 7

Speedup: 1.0×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.75\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.75)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.75)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.75), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 56.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{-4}{\frac{y}{z}}\\ \mathbf{if}\;z \leq -9.1 \cdot 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-213}:\\ \;\;\;\;1 + \frac{4}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+80}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ -4.0 (/ y z)))))
   (if (<= z -9.1e+51)
     t_0
     (if (<= z -9.2e-213) (+ 1.0 (/ 4.0 (/ y x))) (if (<= z 5e+80) 4.0 t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (-4.0 / (y / z));
	double tmp;
	if (z <= -9.1e+51) {
		tmp = t_0;
	} else if (z <= -9.2e-213) {
		tmp = 1.0 + (4.0 / (y / x));
	} else if (z <= 5e+80) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + ((-4.0d0) / (y / z))
    if (z <= (-9.1d+51)) then
        tmp = t_0
    else if (z <= (-9.2d-213)) then
        tmp = 1.0d0 + (4.0d0 / (y / x))
    else if (z <= 5d+80) then
        tmp = 4.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (-4.0 / (y / z));
	double tmp;
	if (z <= -9.1e+51) {
		tmp = t_0;
	} else if (z <= -9.2e-213) {
		tmp = 1.0 + (4.0 / (y / x));
	} else if (z <= 5e+80) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (-4.0 / (y / z))
	tmp = 0
	if z <= -9.1e+51:
		tmp = t_0
	elif z <= -9.2e-213:
		tmp = 1.0 + (4.0 / (y / x))
	elif z <= 5e+80:
		tmp = 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(-4.0 / Float64(y / z)))
	tmp = 0.0
	if (z <= -9.1e+51)
		tmp = t_0;
	elseif (z <= -9.2e-213)
		tmp = Float64(1.0 + Float64(4.0 / Float64(y / x)));
	elseif (z <= 5e+80)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (-4.0 / (y / z));
	tmp = 0.0;
	if (z <= -9.1e+51)
		tmp = t_0;
	elseif (z <= -9.2e-213)
		tmp = 1.0 + (4.0 / (y / x));
	elseif (z <= 5e+80)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(-4.0 / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.1e+51], t$95$0, If[LessEqual[z, -9.2e-213], N[(1.0 + N[(4.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5e+80], 4.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.0999999999999994e51 or 4.99999999999999961e80 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 76.6%

      \[\leadsto 1 + \color{blue}{-4 \cdot \frac{z}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 76.6%

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

      2. associate-/l* 76.5%

         \[\leadsto 1 + \color{blue}{\frac{-4}{\frac{y}{z}}} \]

   4. Simplified 76.5%

      \[\leadsto 1 + \color{blue}{\frac{-4}{\frac{y}{z}}} \]

   if -9.0999999999999994e51 < z < -9.20000000000000011e-213

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 50.8%

      \[\leadsto 1 + \color{blue}{4 \cdot \frac{x}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 50.8%

         \[\leadsto 1 + \color{blue}{\frac{4 \cdot x}{y}} \]

      2. associate-/l* 50.6%

         \[\leadsto 1 + \color{blue}{\frac{4}{\frac{y}{x}}} \]

   4. Simplified 50.6%

      \[\leadsto 1 + \color{blue}{\frac{4}{\frac{y}{x}}} \]

   if -9.20000000000000011e-213 < z < 4.99999999999999961e80

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 60.5%

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

4. Final simplification 64.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.1 \cdot 10^{+51}:\\ \;\;\;\;1 + \frac{-4}{\frac{y}{z}}\\ \mathbf{elif}\;z \leq -9.2 \cdot 10^{-213}:\\ \;\;\;\;1 + \frac{4}{\frac{y}{x}}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{+80}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-4}{\frac{y}{z}}\\ \end{array} \]

Alternative 3: 54.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+86} \lor \neg \left(z \leq 6 \cdot 10^{+81}\right):\\ \;\;\;\;1 + \frac{-4}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.9e+86) (not (<= z 6e+81))) (+ 1.0 (/ -4.0 (/ y z))) 4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.9e+86) || !(z <= 6e+81)) {
		tmp = 1.0 + (-4.0 / (y / z));
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.9d+86)) .or. (.not. (z <= 6d+81))) then
        tmp = 1.0d0 + ((-4.0d0) / (y / z))
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.9e+86) || !(z <= 6e+81)) {
		tmp = 1.0 + (-4.0 / (y / z));
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.9e+86) or not (z <= 6e+81):
		tmp = 1.0 + (-4.0 / (y / z))
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.9e+86) || !(z <= 6e+81))
		tmp = Float64(1.0 + Float64(-4.0 / Float64(y / z)));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.9e+86) || ~((z <= 6e+81)))
		tmp = 1.0 + (-4.0 / (y / z));
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.9e+86], N[Not[LessEqual[z, 6e+81]], $MachinePrecision]], N[(1.0 + N[(-4.0 / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if z < -1.89999999999999989e86 or 5.99999999999999995e81 < z

   1. Initial program 100.0%

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

   2. Taylor expanded in z around inf 80.2%

      \[\leadsto 1 + \color{blue}{-4 \cdot \frac{z}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 80.2%

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

      2. associate-/l* 80.1%

         \[\leadsto 1 + \color{blue}{\frac{-4}{\frac{y}{z}}} \]

   4. Simplified 80.1%

      \[\leadsto 1 + \color{blue}{\frac{-4}{\frac{y}{z}}} \]

   if -1.89999999999999989e86 < z < 5.99999999999999995e81

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 53.1%

      \[\leadsto \color{blue}{4} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+86} \lor \neg \left(z \leq 6 \cdot 10^{+81}\right):\\ \;\;\;\;1 + \frac{-4}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]

Alternative 4: 80.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+36} \lor \neg \left(x \leq 5.8 \cdot 10^{+96}\right):\\ \;\;\;\;1 + \frac{4}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.1e+36) (not (<= x 5.8e+96)))
   (+ 1.0 (/ 4.0 (/ y x)))
   (+ 4.0 (/ (* z -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.1e+36) || !(x <= 5.8e+96)) {
		tmp = 1.0 + (4.0 / (y / x));
	} else {
		tmp = 4.0 + ((z * -4.0) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-3.1d+36)) .or. (.not. (x <= 5.8d+96))) then
        tmp = 1.0d0 + (4.0d0 / (y / x))
    else
        tmp = 4.0d0 + ((z * (-4.0d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.1e+36) || !(x <= 5.8e+96)) {
		tmp = 1.0 + (4.0 / (y / x));
	} else {
		tmp = 4.0 + ((z * -4.0) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -3.1e+36) or not (x <= 5.8e+96):
		tmp = 1.0 + (4.0 / (y / x))
	else:
		tmp = 4.0 + ((z * -4.0) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.1e+36) || !(x <= 5.8e+96))
		tmp = Float64(1.0 + Float64(4.0 / Float64(y / x)));
	else
		tmp = Float64(4.0 + Float64(Float64(z * -4.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -3.1e+36) || ~((x <= 5.8e+96)))
		tmp = 1.0 + (4.0 / (y / x));
	else
		tmp = 4.0 + ((z * -4.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.1e+36], N[Not[LessEqual[x, 5.8e+96]], $MachinePrecision]], N[(1.0 + N[(4.0 / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.0999999999999999e36 or 5.79999999999999955e96 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 72.3%

      \[\leadsto 1 + \color{blue}{4 \cdot \frac{x}{y}} \]
   3. Step-by-step derivation

      1. associate-*r/ 72.3%

         \[\leadsto 1 + \color{blue}{\frac{4 \cdot x}{y}} \]

      2. associate-/l* 72.2%

         \[\leadsto 1 + \color{blue}{\frac{4}{\frac{y}{x}}} \]

   4. Simplified 72.2%

      \[\leadsto 1 + \color{blue}{\frac{4}{\frac{y}{x}}} \]

   if -3.0999999999999999e36 < x < 5.79999999999999955e96

   1. Initial program 99.9%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate--l+ 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. associate-+r+ 99.7%

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

      6. *-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. fma-def 99.7%

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

      9. associate-*r* 99.8%

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

      10. associate-*l/ 99.9%

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

      11. associate-/l* 99.9%

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

      12. *-inverses 99.9%

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

      13. metadata-eval 99.9%

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

      14. metadata-eval 99.9%

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

      15. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 93.0%

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

      1. associate-*r/ 93.0%

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

   6. Simplified 93.0%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{+36} \lor \neg \left(x \leq 5.8 \cdot 10^{+96}\right):\\ \;\;\;\;1 + \frac{4}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \end{array} \]

Alternative 5: 86.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+31} \lor \neg \left(x \leq 8 \cdot 10^{+27}\right):\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1e+31) (not (<= x 8e+27)))
   (+ 4.0 (/ (* 4.0 x) y))
   (+ 4.0 (/ (* z -4.0) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1e+31) || !(x <= 8e+27)) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 + ((z * -4.0) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1d+31)) .or. (.not. (x <= 8d+27))) then
        tmp = 4.0d0 + ((4.0d0 * x) / y)
    else
        tmp = 4.0d0 + ((z * (-4.0d0)) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1e+31) || !(x <= 8e+27)) {
		tmp = 4.0 + ((4.0 * x) / y);
	} else {
		tmp = 4.0 + ((z * -4.0) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1e+31) or not (x <= 8e+27):
		tmp = 4.0 + ((4.0 * x) / y)
	else:
		tmp = 4.0 + ((z * -4.0) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1e+31) || !(x <= 8e+27))
		tmp = Float64(4.0 + Float64(Float64(4.0 * x) / y));
	else
		tmp = Float64(4.0 + Float64(Float64(z * -4.0) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1e+31) || ~((x <= 8e+27)))
		tmp = 4.0 + ((4.0 * x) / y);
	else
		tmp = 4.0 + ((z * -4.0) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1e+31], N[Not[LessEqual[x, 8e+27]], $MachinePrecision]], N[(4.0 + N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -9.9999999999999996e30 or 8.0000000000000001e27 < x

   1. Initial program 99.9%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate--l+ 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. associate-+r+ 99.7%

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

      6. *-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. fma-def 99.7%

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

      9. associate-*r* 99.7%

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

      10. associate-*l/ 99.8%

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

      11. associate-/l* 99.8%

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

      12. *-inverses 99.8%

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

      13. metadata-eval 99.8%

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

      14. metadata-eval 99.8%

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

      15. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 86.6%

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

      1. associate-*r/ 86.6%

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

   6. Simplified 86.6%

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

   if -9.9999999999999996e30 < x < 8.0000000000000001e27

   1. Initial program 99.9%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. associate--l+ 99.7%

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

      4. distribute-lft-in 99.7%

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

      5. associate-+r+ 99.7%

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

      6. *-commutative 99.7%

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

      7. +-commutative 99.7%

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

      8. fma-def 99.7%

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

      9. associate-*r* 99.8%

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

      10. associate-*l/ 99.9%

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

      11. associate-/l* 99.9%

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

      12. *-inverses 99.9%

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

      13. metadata-eval 99.9%

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

      14. metadata-eval 99.9%

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

      15. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 94.4%

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

      1. associate-*r/ 94.4%

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

   6. Simplified 94.4%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+31} \lor \neg \left(x \leq 8 \cdot 10^{+27}\right):\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \end{array} \]

Alternative 6: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-*l/ 99.7%

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

   2. +-commutative 99.7%

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

   3. associate--l+ 99.7%

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

   4. distribute-lft-in 99.7%

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

   5. associate-+r+ 99.7%

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

   6. *-commutative 99.7%

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

   7. +-commutative 99.7%

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

   8. fma-def 99.7%

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

   9. associate-*r* 99.8%

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

   10. associate-*l/ 99.8%

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

   11. associate-/l* 99.8%

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

   12. *-inverses 99.8%

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

   13. metadata-eval 99.8%

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

   14. metadata-eval 99.8%

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

   15. metadata-eval 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 4\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.8%

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

5. Applied egg-rr 99.8%

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

6. Final simplification 99.8%

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

Alternative 7: 34.3% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 4 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 4.0)

C

double code(double x, double y, double z) {
	return 4.0;
}

Fortran

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

Java

public static double code(double x, double y, double z) {
	return 4.0;
}

Python

def code(x, y, z):
	return 4.0

Julia

function code(x, y, z)
	return 4.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 4.0;
end

Wolfram

code[x_, y_, z_] := 4.0

Derivation

1. Initial program 99.9%

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

2. Taylor expanded in y around inf 38.5%

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

3. Final simplification 38.5%

   \[\leadsto 4 \]

Reproduce

herbie shell --seed 2023257 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))