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

Percentage Accurate: 99.7% → 100.0%

Time: 6.1s

Alternatives: 8

Speedup: 1.2×

• 21.2% 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: 99.7% 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: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z) {
	return 1.0 + (4.0 * (0.75 + ((x - 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 * (0.75d0 + ((x - z) / y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. associate-/l* 99.9%

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

   2. associate--l+ 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in x around 0 96.9%

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

   1. associate--l+ 96.9%

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

   2. div-sub 100.0%

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

7. Simplified 100.0%

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

Alternative 2: 79.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+141} \lor \neg \left(z \leq 9.6 \cdot 10^{-32} \lor \neg \left(z \leq 3.9 \cdot 10^{+18}\right) \land z \leq 3.05 \cdot 10^{+116}\right):\\ \;\;\;\;1 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \left(0.75 + \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -7e+141)
         (not (or (<= z 9.6e-32) (and (not (<= z 3.9e+18)) (<= z 3.05e+116)))))
   (+ 1.0 (/ (* z -4.0) y))
   (+ 1.0 (* 4.0 (+ 0.75 (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7e+141) || !((z <= 9.6e-32) || (!(z <= 3.9e+18) && (z <= 3.05e+116)))) {
		tmp = 1.0 + ((z * -4.0) / y);
	} else {
		tmp = 1.0 + (4.0 * (0.75 + (x / 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 ((z <= (-7d+141)) .or. (.not. (z <= 9.6d-32) .or. (.not. (z <= 3.9d+18)) .and. (z <= 3.05d+116))) then
        tmp = 1.0d0 + ((z * (-4.0d0)) / y)
    else
        tmp = 1.0d0 + (4.0d0 * (0.75d0 + (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -7e+141) || !((z <= 9.6e-32) || (!(z <= 3.9e+18) && (z <= 3.05e+116)))) {
		tmp = 1.0 + ((z * -4.0) / y);
	} else {
		tmp = 1.0 + (4.0 * (0.75 + (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -7e+141) or not ((z <= 9.6e-32) or (not (z <= 3.9e+18) and (z <= 3.05e+116))):
		tmp = 1.0 + ((z * -4.0) / y)
	else:
		tmp = 1.0 + (4.0 * (0.75 + (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -7e+141) || !((z <= 9.6e-32) || (!(z <= 3.9e+18) && (z <= 3.05e+116))))
		tmp = Float64(1.0 + Float64(Float64(z * -4.0) / y));
	else
		tmp = Float64(1.0 + Float64(4.0 * Float64(0.75 + Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -7e+141) || ~(((z <= 9.6e-32) || (~((z <= 3.9e+18)) && (z <= 3.05e+116)))))
		tmp = 1.0 + ((z * -4.0) / y);
	else
		tmp = 1.0 + (4.0 * (0.75 + (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -7e+141], N[Not[Or[LessEqual[z, 9.6e-32], And[N[Not[LessEqual[z, 3.9e+18]], $MachinePrecision], LessEqual[z, 3.05e+116]]]], $MachinePrecision]], N[(1.0 + N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(4.0 * N[(0.75 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.9999999999999999e141 or 9.6000000000000005e-32 < z < 3.9e18 or 3.05000000000000009e116 < z

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 79.2%

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

      1. neg-mul-1 79.2%

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

      2. distribute-neg-frac2 79.2%

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

   7. Simplified 79.2%

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

   8. Taylor expanded in z around 0 79.2%

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

      1. associate-*r/ 79.2%

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

   10. Simplified 79.2%

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

   if -6.9999999999999999e141 < z < 9.6000000000000005e-32 or 3.9e18 < z < 3.05000000000000009e116

   1. Initial program 99.3%

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

      1. associate-/l* 99.9%

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

      2. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in z around 0 87.2%

      \[\leadsto 1 + 4 \cdot \color{blue}{\left(0.75 + \frac{x}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+141} \lor \neg \left(z \leq 9.6 \cdot 10^{-32} \lor \neg \left(z \leq 3.9 \cdot 10^{+18}\right) \land z \leq 3.05 \cdot 10^{+116}\right):\\ \;\;\;\;1 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \left(0.75 + \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+50} \lor \neg \left(y \leq 1.25 \cdot 10^{+87}\right):\\ \;\;\;\;1 + 4 \cdot \left(0.75 + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \frac{x - z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.9e+50) (not (<= y 1.25e+87)))
   (+ 1.0 (* 4.0 (+ 0.75 (/ x y))))
   (+ 1.0 (* 4.0 (/ (- x z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.9e+50) || !(y <= 1.25e+87)) {
		tmp = 1.0 + (4.0 * (0.75 + (x / y)));
	} else {
		tmp = 1.0 + (4.0 * ((x - z) / 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 ((y <= (-2.9d+50)) .or. (.not. (y <= 1.25d+87))) then
        tmp = 1.0d0 + (4.0d0 * (0.75d0 + (x / y)))
    else
        tmp = 1.0d0 + (4.0d0 * ((x - z) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.9e+50) || !(y <= 1.25e+87)) {
		tmp = 1.0 + (4.0 * (0.75 + (x / y)));
	} else {
		tmp = 1.0 + (4.0 * ((x - z) / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -2.9e+50) or not (y <= 1.25e+87):
		tmp = 1.0 + (4.0 * (0.75 + (x / y)))
	else:
		tmp = 1.0 + (4.0 * ((x - z) / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.9e+50) || !(y <= 1.25e+87))
		tmp = Float64(1.0 + Float64(4.0 * Float64(0.75 + Float64(x / y))));
	else
		tmp = Float64(1.0 + Float64(4.0 * Float64(Float64(x - z) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -2.9e+50) || ~((y <= 1.25e+87)))
		tmp = 1.0 + (4.0 * (0.75 + (x / y)));
	else
		tmp = 1.0 + (4.0 * ((x - z) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.9e+50], N[Not[LessEqual[y, 1.25e+87]], $MachinePrecision]], N[(1.0 + N[(4.0 * N[(0.75 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.9e50 or 1.24999999999999995e87 < y

   1. Initial program 98.9%

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

      1. associate-/l* 99.9%

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

      2. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in z around 0 87.3%

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

   if -2.9e50 < y < 1.24999999999999995e87

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 94.0%

      \[\leadsto 1 + 4 \cdot \color{blue}{\frac{x - z}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+50} \lor \neg \left(y \leq 1.25 \cdot 10^{+87}\right):\\ \;\;\;\;1 + 4 \cdot \left(0.75 + \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \frac{x - z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4400000000 \lor \neg \left(z \leq 10^{-79}\right):\\ \;\;\;\;1 + 4 \cdot \left(0.75 - \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \left(0.75 + \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -4400000000.0) (not (<= z 1e-79)))
   (+ 1.0 (* 4.0 (- 0.75 (/ z y))))
   (+ 1.0 (* 4.0 (+ 0.75 (/ x y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4400000000.0) || !(z <= 1e-79)) {
		tmp = 1.0 + (4.0 * (0.75 - (z / y)));
	} else {
		tmp = 1.0 + (4.0 * (0.75 + (x / 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 ((z <= (-4400000000.0d0)) .or. (.not. (z <= 1d-79))) then
        tmp = 1.0d0 + (4.0d0 * (0.75d0 - (z / y)))
    else
        tmp = 1.0d0 + (4.0d0 * (0.75d0 + (x / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -4400000000.0) || !(z <= 1e-79)) {
		tmp = 1.0 + (4.0 * (0.75 - (z / y)));
	} else {
		tmp = 1.0 + (4.0 * (0.75 + (x / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -4400000000.0) or not (z <= 1e-79):
		tmp = 1.0 + (4.0 * (0.75 - (z / y)))
	else:
		tmp = 1.0 + (4.0 * (0.75 + (x / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -4400000000.0) || !(z <= 1e-79))
		tmp = Float64(1.0 + Float64(4.0 * Float64(0.75 - Float64(z / y))));
	else
		tmp = Float64(1.0 + Float64(4.0 * Float64(0.75 + Float64(x / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -4400000000.0) || ~((z <= 1e-79)))
		tmp = 1.0 + (4.0 * (0.75 - (z / y)));
	else
		tmp = 1.0 + (4.0 * (0.75 + (x / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -4400000000.0], N[Not[LessEqual[z, 1e-79]], $MachinePrecision]], N[(1.0 + N[(4.0 * N[(0.75 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(4.0 * N[(0.75 + N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4e9 or 1e-79 < z

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 82.7%

      \[\leadsto 1 + 4 \cdot \color{blue}{\frac{0.75 \cdot y - z}{y}} \]
   6. Step-by-step derivation

      1. div-sub 82.7%

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

      2. associate-/l* 82.7%

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

      3. *-inverses 82.7%

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

      4. metadata-eval 82.7%

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

   7. Simplified 82.7%

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

   if -4.4e9 < z < 1e-79

   1. Initial program 99.0%

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

      1. associate-/l* 99.9%

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

      2. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around 0 100.0%

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

   6. Taylor expanded in z around 0 98.1%

      \[\leadsto 1 + 4 \cdot \color{blue}{\left(0.75 + \frac{x}{y}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4400000000 \lor \neg \left(z \leq 10^{-79}\right):\\ \;\;\;\;1 + 4 \cdot \left(0.75 - \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \left(0.75 + \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -120000000000 \lor \neg \left(z \leq 4 \cdot 10^{-80}\right):\\ \;\;\;\;1 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -120000000000.0) (not (<= z 4e-80)))
   (+ 1.0 (/ (* z -4.0) y))
   (+ 1.0 (* 4.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -120000000000.0) || !(z <= 4e-80)) {
		tmp = 1.0 + ((z * -4.0) / y);
	} else {
		tmp = 1.0 + (4.0 * (x / 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 ((z <= (-120000000000.0d0)) .or. (.not. (z <= 4d-80))) then
        tmp = 1.0d0 + ((z * (-4.0d0)) / y)
    else
        tmp = 1.0d0 + (4.0d0 * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -120000000000.0) || !(z <= 4e-80)) {
		tmp = 1.0 + ((z * -4.0) / y);
	} else {
		tmp = 1.0 + (4.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -120000000000.0) or not (z <= 4e-80):
		tmp = 1.0 + ((z * -4.0) / y)
	else:
		tmp = 1.0 + (4.0 * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -120000000000.0) || !(z <= 4e-80))
		tmp = Float64(1.0 + Float64(Float64(z * -4.0) / y));
	else
		tmp = Float64(1.0 + Float64(4.0 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -120000000000.0) || ~((z <= 4e-80)))
		tmp = 1.0 + ((z * -4.0) / y);
	else
		tmp = 1.0 + (4.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -120000000000.0], N[Not[LessEqual[z, 4e-80]], $MachinePrecision]], N[(1.0 + N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e11 or 3.99999999999999985e-80 < z

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 64.4%

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

      1. neg-mul-1 64.4%

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

      2. distribute-neg-frac2 64.4%

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

   7. Simplified 64.4%

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

   8. Taylor expanded in z around 0 64.4%

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

      1. associate-*r/ 64.4%

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

   10. Simplified 64.4%

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

   if -1.2e11 < z < 3.99999999999999985e-80

   1. Initial program 99.0%

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

      1. associate-/l* 99.9%

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

      2. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 61.5%

      \[\leadsto 1 + 4 \cdot \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -120000000000 \lor \neg \left(z \leq 4 \cdot 10^{-80}\right):\\ \;\;\;\;1 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -13500000000000 \lor \neg \left(z \leq 10^{-79}\right):\\ \;\;\;\;1 + z \cdot \frac{-4}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -13500000000000.0) (not (<= z 1e-79)))
   (+ 1.0 (* z (/ -4.0 y)))
   (+ 1.0 (* 4.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -13500000000000.0) || !(z <= 1e-79)) {
		tmp = 1.0 + (z * (-4.0 / y));
	} else {
		tmp = 1.0 + (4.0 * (x / 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 ((z <= (-13500000000000.0d0)) .or. (.not. (z <= 1d-79))) then
        tmp = 1.0d0 + (z * ((-4.0d0) / y))
    else
        tmp = 1.0d0 + (4.0d0 * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -13500000000000.0) || !(z <= 1e-79)) {
		tmp = 1.0 + (z * (-4.0 / y));
	} else {
		tmp = 1.0 + (4.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -13500000000000.0) or not (z <= 1e-79):
		tmp = 1.0 + (z * (-4.0 / y))
	else:
		tmp = 1.0 + (4.0 * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -13500000000000.0) || !(z <= 1e-79))
		tmp = Float64(1.0 + Float64(z * Float64(-4.0 / y)));
	else
		tmp = Float64(1.0 + Float64(4.0 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -13500000000000.0) || ~((z <= 1e-79)))
		tmp = 1.0 + (z * (-4.0 / y));
	else
		tmp = 1.0 + (4.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -13500000000000.0], N[Not[LessEqual[z, 1e-79]], $MachinePrecision]], N[(1.0 + N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.35e13 or 1e-79 < z

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around inf 64.4%

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

      1. neg-mul-1 64.4%

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

      2. distribute-neg-frac2 64.4%

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

   7. Simplified 64.4%

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

   8. Taylor expanded in z around 0 64.4%

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

      1. *-commutative 64.4%

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

      2. associate-*l/ 64.4%

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

      3. associate-/l* 64.2%

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

   10. Simplified 64.2%

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

   if -1.35e13 < z < 1e-79

   1. Initial program 99.0%

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

      1. associate-/l* 99.9%

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

      2. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 61.5%

      \[\leadsto 1 + 4 \cdot \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -13500000000000 \lor \neg \left(z \leq 10^{-79}\right):\\ \;\;\;\;1 + z \cdot \frac{-4}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+66}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+122}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -3.2e+66) 4.0 (if (<= y 3.3e+122) (+ 1.0 (* 4.0 (/ x y))) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e+66) {
		tmp = 4.0;
	} else if (y <= 3.3e+122) {
		tmp = 1.0 + (4.0 * (x / y));
	} 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 (y <= (-3.2d+66)) then
        tmp = 4.0d0
    else if (y <= 3.3d+122) then
        tmp = 1.0d0 + (4.0d0 * (x / y))
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -3.2e+66) {
		tmp = 4.0;
	} else if (y <= 3.3e+122) {
		tmp = 1.0 + (4.0 * (x / y));
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -3.2e+66:
		tmp = 4.0
	elif y <= 3.3e+122:
		tmp = 1.0 + (4.0 * (x / y))
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -3.2e+66)
		tmp = 4.0;
	elseif (y <= 3.3e+122)
		tmp = Float64(1.0 + Float64(4.0 * Float64(x / y)));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -3.2e+66)
		tmp = 4.0;
	elseif (y <= 3.3e+122)
		tmp = 1.0 + (4.0 * (x / y));
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -3.2e+66], 4.0, If[LessEqual[y, 3.3e+122], N[(1.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -3.2e66 or 3.2999999999999999e122 < y

   1. Initial program 98.8%

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

      1. associate-/l* 99.9%

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

      2. associate--l+ 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around inf 72.9%

      \[\leadsto 1 + 4 \cdot \color{blue}{0.75} \]

   if -3.2e66 < y < 3.2999999999999999e122

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

      2. associate--l+ 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 48.3%

      \[\leadsto 1 + 4 \cdot \color{blue}{\frac{x}{y}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+66}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 3.3 \cdot 10^{+122}:\\ \;\;\;\;1 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.0% 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 99.6%

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

   1. associate-/l* 99.9%

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

   2. associate--l+ 99.9%

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

3. Simplified 99.9%

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

5. Taylor expanded in y around inf 30.9%

   \[\leadsto 1 + 4 \cdot \color{blue}{0.75} \]

6. Final simplification 30.9%

   \[\leadsto 4 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024091 
(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)))