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

Percentage Accurate: 99.7% → 99.8%

Time: 6.9s

Alternatives: 11

Speedup: 1.2×

• 21.5% 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: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ 1 + \left(\frac{4 \cdot \left(x - z\right)}{y} + 3\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ 1.0 (+ (/ (* 4.0 (- x z)) y) 3.0)))

C

double code(double x, double y, double z) {
	return 1.0 + (((4.0 * (x - z)) / y) + 3.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + (((4.0d0 * (x - z)) / y) + 3.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + (((4.0 * (x - z)) / y) + 3.0);
}

Python

def code(x, y, z):
	return 1.0 + (((4.0 * (x - z)) / y) + 3.0)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(Float64(4.0 * Float64(x - z)) / y) + 3.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + (((4.0 * (x - z)) / y) + 3.0);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(N[(4.0 * N[(x - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] + 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-*r/ 100.0%

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

5. Simplified 100.0%

   \[\leadsto 1 + \color{blue}{\left(\frac{4 \cdot \left(x - z\right)}{y} + 3\right)} \]
6. Add Preprocessing

Alternative 2: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{+61} \lor \neg \left(y \leq 1.4 \cdot 10^{+71}\right):\\ \;\;\;\;1 + \left(3 + -4 \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{4 \cdot \left(x - z\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -9.8e+61) (not (<= y 1.4e+71)))
   (+ 1.0 (+ 3.0 (* -4.0 (/ z y))))
   (+ 1.0 (/ (* 4.0 (- x z)) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -9.8e+61) || !(y <= 1.4e+71)) {
		tmp = 1.0 + (3.0 + (-4.0 * (z / 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 <= (-9.8d+61)) .or. (.not. (y <= 1.4d+71))) then
        tmp = 1.0d0 + (3.0d0 + ((-4.0d0) * (z / 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 <= -9.8e+61) || !(y <= 1.4e+71)) {
		tmp = 1.0 + (3.0 + (-4.0 * (z / y)));
	} else {
		tmp = 1.0 + ((4.0 * (x - z)) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -9.8e+61) or not (y <= 1.4e+71):
		tmp = 1.0 + (3.0 + (-4.0 * (z / y)))
	else:
		tmp = 1.0 + ((4.0 * (x - z)) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -9.8e+61) || !(y <= 1.4e+71))
		tmp = Float64(1.0 + Float64(3.0 + Float64(-4.0 * Float64(z / y))));
	else
		tmp = Float64(1.0 + Float64(Float64(4.0 * Float64(x - z)) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -9.8e+61) || ~((y <= 1.4e+71)))
		tmp = 1.0 + (3.0 + (-4.0 * (z / y)));
	else
		tmp = 1.0 + ((4.0 * (x - z)) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -9.8e+61], N[Not[LessEqual[y, 1.4e+71]], $MachinePrecision]], N[(1.0 + N[(3.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(4.0 * N[(x - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.8000000000000005e61 or 1.40000000000000001e71 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 89.9%

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

   if -9.8000000000000005e61 < y < 1.40000000000000001e71

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf 91.4%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.8 \cdot 10^{+61} \lor \neg \left(y \leq 1.4 \cdot 10^{+71}\right):\\ \;\;\;\;1 + \left(3 + -4 \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{4 \cdot \left(x - z\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+46} \lor \neg \left(x \leq 1.35 \cdot 10^{-104}\right):\\ \;\;\;\;\left(x + y\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(3 + -4 \cdot \frac{z}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -4.6e+46) (not (<= x 1.35e-104)))
   (* (+ x y) (/ 4.0 y))
   (+ 1.0 (+ 3.0 (* -4.0 (/ z y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.6e+46) || !(x <= 1.35e-104)) {
		tmp = (x + y) * (4.0 / y);
	} else {
		tmp = 1.0 + (3.0 + (-4.0 * (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 ((x <= (-4.6d+46)) .or. (.not. (x <= 1.35d-104))) then
        tmp = (x + y) * (4.0d0 / y)
    else
        tmp = 1.0d0 + (3.0d0 + ((-4.0d0) * (z / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -4.6e+46) || !(x <= 1.35e-104)) {
		tmp = (x + y) * (4.0 / y);
	} else {
		tmp = 1.0 + (3.0 + (-4.0 * (z / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -4.6e+46) or not (x <= 1.35e-104):
		tmp = (x + y) * (4.0 / y)
	else:
		tmp = 1.0 + (3.0 + (-4.0 * (z / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -4.6e+46) || !(x <= 1.35e-104))
		tmp = Float64(Float64(x + y) * Float64(4.0 / y));
	else
		tmp = Float64(1.0 + Float64(3.0 + Float64(-4.0 * Float64(z / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -4.6e+46) || ~((x <= 1.35e-104)))
		tmp = (x + y) * (4.0 / y);
	else
		tmp = 1.0 + (3.0 + (-4.0 * (z / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -4.6e+46], N[Not[LessEqual[x, 1.35e-104]], $MachinePrecision]], N[(N[(x + y), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(3.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.6000000000000001e46 or 1.3499999999999999e-104 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 83.9%

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

      1. *-commutative 83.9%

         \[\leadsto 1 + \left(\color{blue}{\frac{x}{y} \cdot 4} + 3\right) \]

      2. associate-*l/ 83.9%

         \[\leadsto 1 + \left(\color{blue}{\frac{x \cdot 4}{y}} + 3\right) \]

      3. associate-*r/ 83.8%

         \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

   8. Simplified 83.8%

      \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

   9. Taylor expanded in y around 0 83.9%

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

       1. distribute-lft-out 83.9%

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

       2. *-commutative 83.9%

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

       3. associate-/l* 83.8%

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

   11. Simplified 83.8%

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

   if -4.6000000000000001e46 < x < 1.3499999999999999e-104

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 94.6%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.6 \cdot 10^{+46} \lor \neg \left(x \leq 1.35 \cdot 10^{-104}\right):\\ \;\;\;\;\left(x + y\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(3 + -4 \cdot \frac{z}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+47} \lor \neg \left(x \leq 1.35 \cdot 10^{-104}\right):\\ \;\;\;\;\left(x + y\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(3 + z \cdot \frac{-4}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2e+47) (not (<= x 1.35e-104)))
   (* (+ x y) (/ 4.0 y))
   (+ 1.0 (+ 3.0 (* z (/ -4.0 y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2e+47) || !(x <= 1.35e-104)) {
		tmp = (x + y) * (4.0 / y);
	} else {
		tmp = 1.0 + (3.0 + (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 <= (-2d+47)) .or. (.not. (x <= 1.35d-104))) then
        tmp = (x + y) * (4.0d0 / y)
    else
        tmp = 1.0d0 + (3.0d0 + (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 <= -2e+47) || !(x <= 1.35e-104)) {
		tmp = (x + y) * (4.0 / y);
	} else {
		tmp = 1.0 + (3.0 + (z * (-4.0 / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -2e+47) or not (x <= 1.35e-104):
		tmp = (x + y) * (4.0 / y)
	else:
		tmp = 1.0 + (3.0 + (z * (-4.0 / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2e+47) || !(x <= 1.35e-104))
		tmp = Float64(Float64(x + y) * Float64(4.0 / y));
	else
		tmp = Float64(1.0 + Float64(3.0 + Float64(z * Float64(-4.0 / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -2e+47) || ~((x <= 1.35e-104)))
		tmp = (x + y) * (4.0 / y);
	else
		tmp = 1.0 + (3.0 + (z * (-4.0 / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2e+47], N[Not[LessEqual[x, 1.35e-104]], $MachinePrecision]], N[(N[(x + y), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(3.0 + N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.0000000000000001e47 or 1.3499999999999999e-104 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 83.9%

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

      1. *-commutative 83.9%

         \[\leadsto 1 + \left(\color{blue}{\frac{x}{y} \cdot 4} + 3\right) \]

      2. associate-*l/ 83.9%

         \[\leadsto 1 + \left(\color{blue}{\frac{x \cdot 4}{y}} + 3\right) \]

      3. associate-*r/ 83.8%

         \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

   8. Simplified 83.8%

      \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

   9. Taylor expanded in y around 0 83.9%

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

       1. distribute-lft-out 83.9%

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

       2. *-commutative 83.9%

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

       3. associate-/l* 83.8%

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

   11. Simplified 83.8%

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

   if -2.0000000000000001e47 < x < 1.3499999999999999e-104

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 94.5%

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

      1. div-sub 94.5%

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

      2. associate-/l* 94.6%

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

      3. *-inverses 94.6%

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

      4. metadata-eval 94.6%

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

      5. sub-neg 94.6%

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

      6. distribute-lft-in 94.6%

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

      7. metadata-eval 94.6%

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

      8. distribute-rgt-neg-in 94.6%

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

      9. *-lft-identity 94.6%

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

      10. associate-*l/ 94.4%

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

      11. associate-*l* 94.4%

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

      12. *-commutative 94.4%

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

      13. distribute-rgt-neg-in 94.4%

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

      14. associate-*r/ 94.4%

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

      15. metadata-eval 94.4%

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

      16. distribute-neg-frac 94.4%

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

      17. metadata-eval 94.4%

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

   5. Simplified 94.4%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{+47} \lor \neg \left(x \leq 1.35 \cdot 10^{-104}\right):\\ \;\;\;\;\left(x + y\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(3 + z \cdot \frac{-4}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+46}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{4}{y}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-106}:\\ \;\;\;\;1 + \left(3 + -4 \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(3 + x \cdot \frac{4}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -2.5e+46)
   (* (+ x y) (/ 4.0 y))
   (if (<= x 1.75e-106)
     (+ 1.0 (+ 3.0 (* -4.0 (/ z y))))
     (+ 1.0 (+ 3.0 (* x (/ 4.0 y)))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.5e+46) {
		tmp = (x + y) * (4.0 / y);
	} else if (x <= 1.75e-106) {
		tmp = 1.0 + (3.0 + (-4.0 * (z / y)));
	} else {
		tmp = 1.0 + (3.0 + (x * (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 <= (-2.5d+46)) then
        tmp = (x + y) * (4.0d0 / y)
    else if (x <= 1.75d-106) then
        tmp = 1.0d0 + (3.0d0 + ((-4.0d0) * (z / y)))
    else
        tmp = 1.0d0 + (3.0d0 + (x * (4.0d0 / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -2.5e+46) {
		tmp = (x + y) * (4.0 / y);
	} else if (x <= 1.75e-106) {
		tmp = 1.0 + (3.0 + (-4.0 * (z / y)));
	} else {
		tmp = 1.0 + (3.0 + (x * (4.0 / y)));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -2.5e+46:
		tmp = (x + y) * (4.0 / y)
	elif x <= 1.75e-106:
		tmp = 1.0 + (3.0 + (-4.0 * (z / y)))
	else:
		tmp = 1.0 + (3.0 + (x * (4.0 / y)))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -2.5e+46)
		tmp = Float64(Float64(x + y) * Float64(4.0 / y));
	elseif (x <= 1.75e-106)
		tmp = Float64(1.0 + Float64(3.0 + Float64(-4.0 * Float64(z / y))));
	else
		tmp = Float64(1.0 + Float64(3.0 + Float64(x * Float64(4.0 / y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -2.5e+46)
		tmp = (x + y) * (4.0 / y);
	elseif (x <= 1.75e-106)
		tmp = 1.0 + (3.0 + (-4.0 * (z / y)));
	else
		tmp = 1.0 + (3.0 + (x * (4.0 / y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -2.5e+46], N[(N[(x + y), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.75e-106], N[(1.0 + N[(3.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(3.0 + N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.5000000000000001e46

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 84.2%

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

      1. *-commutative 84.2%

         \[\leadsto 1 + \left(\color{blue}{\frac{x}{y} \cdot 4} + 3\right) \]

      2. associate-*l/ 84.2%

         \[\leadsto 1 + \left(\color{blue}{\frac{x \cdot 4}{y}} + 3\right) \]

      3. associate-*r/ 84.1%

         \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

   8. Simplified 84.1%

      \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

   9. Taylor expanded in y around 0 84.2%

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

       1. distribute-lft-out 84.2%

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

       2. *-commutative 84.2%

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

       3. associate-/l* 84.1%

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

   11. Simplified 84.1%

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

   if -2.5000000000000001e46 < x < 1.75e-106

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around 0 94.5%

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

   if 1.75e-106 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 84.1%

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

      1. *-commutative 84.1%

         \[\leadsto 1 + \left(\color{blue}{\frac{x}{y} \cdot 4} + 3\right) \]

      2. associate-*l/ 84.1%

         \[\leadsto 1 + \left(\color{blue}{\frac{x \cdot 4}{y}} + 3\right) \]

      3. associate-*r/ 84.0%

         \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

   8. Simplified 84.0%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.5 \cdot 10^{+46}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{4}{y}\\ \mathbf{elif}\;x \leq 1.75 \cdot 10^{-106}:\\ \;\;\;\;1 + \left(3 + -4 \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;1 + \left(3 + x \cdot \frac{4}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+244} \lor \neg \left(z \leq 6.6 \cdot 10^{+209}\right):\\ \;\;\;\;1 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.75e+244) (not (<= z 6.6e+209)))
   (+ 1.0 (/ (* z -4.0) y))
   (* (+ x y) (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.75e+244) || !(z <= 6.6e+209)) {
		tmp = 1.0 + ((z * -4.0) / y);
	} else {
		tmp = (x + y) * (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 ((z <= (-1.75d+244)) .or. (.not. (z <= 6.6d+209))) then
        tmp = 1.0d0 + ((z * (-4.0d0)) / y)
    else
        tmp = (x + y) * (4.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.75e+244) || !(z <= 6.6e+209)) {
		tmp = 1.0 + ((z * -4.0) / y);
	} else {
		tmp = (x + y) * (4.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.75e+244) or not (z <= 6.6e+209):
		tmp = 1.0 + ((z * -4.0) / y)
	else:
		tmp = (x + y) * (4.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.75e+244) || !(z <= 6.6e+209))
		tmp = Float64(1.0 + Float64(Float64(z * -4.0) / y));
	else
		tmp = Float64(Float64(x + y) * Float64(4.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.75e+244) || ~((z <= 6.6e+209)))
		tmp = 1.0 + ((z * -4.0) / y);
	else
		tmp = (x + y) * (4.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.75e+244], N[Not[LessEqual[z, 6.6e+209]], $MachinePrecision]], N[(1.0 + N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.74999999999999987e244 or 6.59999999999999961e209 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 85.3%

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

      1. *-commutative 85.3%

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

      2. associate-*l/ 85.3%

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

   5. Simplified 85.3%

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

   if -1.74999999999999987e244 < z < 6.59999999999999961e209

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 77.4%

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

      1. *-commutative 77.4%

         \[\leadsto 1 + \left(\color{blue}{\frac{x}{y} \cdot 4} + 3\right) \]

      2. associate-*l/ 77.4%

         \[\leadsto 1 + \left(\color{blue}{\frac{x \cdot 4}{y}} + 3\right) \]

      3. associate-*r/ 76.9%

         \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

   8. Simplified 76.9%

      \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

   9. Taylor expanded in y around 0 76.9%

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

       1. distribute-lft-out 76.9%

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

       2. *-commutative 76.9%

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

       3. associate-/l* 76.8%

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

   11. Simplified 76.8%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+244} \lor \neg \left(z \leq 6.6 \cdot 10^{+209}\right):\\ \;\;\;\;1 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+244} \lor \neg \left(z \leq 6.1 \cdot 10^{+209}\right):\\ \;\;\;\;1 + z \cdot \frac{-4}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.75e+244) (not (<= z 6.1e+209)))
   (+ 1.0 (* z (/ -4.0 y)))
   (* (+ x y) (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.75e+244) || !(z <= 6.1e+209)) {
		tmp = 1.0 + (z * (-4.0 / y));
	} else {
		tmp = (x + y) * (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 ((z <= (-1.75d+244)) .or. (.not. (z <= 6.1d+209))) then
        tmp = 1.0d0 + (z * ((-4.0d0) / y))
    else
        tmp = (x + y) * (4.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.75e+244) || !(z <= 6.1e+209)) {
		tmp = 1.0 + (z * (-4.0 / y));
	} else {
		tmp = (x + y) * (4.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.75e+244) or not (z <= 6.1e+209):
		tmp = 1.0 + (z * (-4.0 / y))
	else:
		tmp = (x + y) * (4.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.75e+244) || !(z <= 6.1e+209))
		tmp = Float64(1.0 + Float64(z * Float64(-4.0 / y)));
	else
		tmp = Float64(Float64(x + y) * Float64(4.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.75e+244) || ~((z <= 6.1e+209)))
		tmp = 1.0 + (z * (-4.0 / y));
	else
		tmp = (x + y) * (4.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.75e+244], N[Not[LessEqual[z, 6.1e+209]], $MachinePrecision]], N[(1.0 + N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + y), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.74999999999999987e244 or 6.1000000000000003e209 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf 85.3%

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

      1. metadata-eval 85.3%

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

      2. distribute-lft-neg-in 85.3%

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

      3. *-lft-identity 85.3%

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

      4. associate-*l/ 85.1%

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

      5. associate-*l* 85.1%

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

      6. *-commutative 85.1%

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

      7. distribute-rgt-neg-in 85.1%

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

      8. associate-*r/ 85.1%

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

      9. metadata-eval 85.1%

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

      10. distribute-neg-frac 85.1%

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

      11. metadata-eval 85.1%

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

   5. Simplified 85.1%

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

   if -1.74999999999999987e244 < z < 6.1000000000000003e209

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 77.4%

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

      1. *-commutative 77.4%

         \[\leadsto 1 + \left(\color{blue}{\frac{x}{y} \cdot 4} + 3\right) \]

      2. associate-*l/ 77.4%

         \[\leadsto 1 + \left(\color{blue}{\frac{x \cdot 4}{y}} + 3\right) \]

      3. associate-*r/ 76.9%

         \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

   8. Simplified 76.9%

      \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

   9. Taylor expanded in y around 0 76.9%

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

       1. distribute-lft-out 76.9%

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

       2. *-commutative 76.9%

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

       3. associate-/l* 76.8%

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

   11. Simplified 76.8%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{+244} \lor \neg \left(z \leq 6.1 \cdot 10^{+209}\right):\\ \;\;\;\;1 + z \cdot \frac{-4}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x + y\right) \cdot \frac{4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 53.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.75 \cdot 10^{+58}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{+91}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -4.75e+58) 4.0 (if (<= y 2.4e+91) (* 4.0 (/ x y)) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -4.75e+58) {
		tmp = 4.0;
	} else if (y <= 2.4e+91) {
		tmp = 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 <= (-4.75d+58)) then
        tmp = 4.0d0
    else if (y <= 2.4d+91) then
        tmp = 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 <= -4.75e+58) {
		tmp = 4.0;
	} else if (y <= 2.4e+91) {
		tmp = 4.0 * (x / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -4.75e+58:
		tmp = 4.0
	elif y <= 2.4e+91:
		tmp = 4.0 * (x / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -4.75e+58)
		tmp = 4.0;
	elseif (y <= 2.4e+91)
		tmp = 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 <= -4.75e+58)
		tmp = 4.0;
	elseif (y <= 2.4e+91)
		tmp = 4.0 * (x / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -4.75e+58], 4.0, If[LessEqual[y, 2.4e+91], N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -4.7500000000000001e58 or 2.39999999999999983e91 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 83.5%

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

      1. *-commutative 83.5%

         \[\leadsto 1 + \left(\color{blue}{\frac{x}{y} \cdot 4} + 3\right) \]

      2. associate-*l/ 83.5%

         \[\leadsto 1 + \left(\color{blue}{\frac{x \cdot 4}{y}} + 3\right) \]

      3. associate-*r/ 83.4%

         \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

   8. Simplified 83.4%

      \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

   9. Taylor expanded in x around 0 74.6%

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

   if -4.7500000000000001e58 < y < 2.39999999999999983e91

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*r/ 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 58.5%

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

      1. *-commutative 58.5%

         \[\leadsto 1 + \left(\color{blue}{\frac{x}{y} \cdot 4} + 3\right) \]

      2. associate-*l/ 58.5%

         \[\leadsto 1 + \left(\color{blue}{\frac{x \cdot 4}{y}} + 3\right) \]

      3. associate-*r/ 57.8%

         \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

   8. Simplified 57.8%

      \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

   9. Taylor expanded in x around inf 49.0%

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

Alternative 9: 67.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in y around inf 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-*r/ 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around inf 68.5%

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

   1. *-commutative 68.5%

      \[\leadsto 1 + \left(\color{blue}{\frac{x}{y} \cdot 4} + 3\right) \]

   2. associate-*l/ 68.5%

      \[\leadsto 1 + \left(\color{blue}{\frac{x \cdot 4}{y}} + 3\right) \]

   3. associate-*r/ 68.0%

      \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

8. Simplified 68.0%

   \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

9. Taylor expanded in y around 0 68.1%

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

    1. distribute-lft-out 68.1%

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

    2. *-commutative 68.1%

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

    3. associate-/l* 67.9%

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

11. Simplified 67.9%

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

Alternative 10: 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 99.9%

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

3. Taylor expanded in y around inf 100.0%

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

   1. +-commutative 100.0%

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

   2. associate-*r/ 100.0%

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

5. Simplified 100.0%

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

6. Taylor expanded in x around inf 68.5%

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

   1. *-commutative 68.5%

      \[\leadsto 1 + \left(\color{blue}{\frac{x}{y} \cdot 4} + 3\right) \]

   2. associate-*l/ 68.5%

      \[\leadsto 1 + \left(\color{blue}{\frac{x \cdot 4}{y}} + 3\right) \]

   3. associate-*r/ 68.0%

      \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

8. Simplified 68.0%

   \[\leadsto 1 + \left(\color{blue}{x \cdot \frac{4}{y}} + 3\right) \]

9. Taylor expanded in x around 0 36.8%

   \[\leadsto \color{blue}{4} \]
10. Add Preprocessing

Alternative 11: 7.6% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 1.0)

C

double code(double x, double y, double z) {
	return 1.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 1.0;
}

Python

def code(x, y, z):
	return 1.0

Julia

function code(x, y, z)
	return 1.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0;
end

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in z around inf 38.7%

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

   1. metadata-eval 38.7%

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

   2. distribute-lft-neg-in 38.7%

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

   3. *-lft-identity 38.7%

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

   4. associate-*l/ 38.6%

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

   5. associate-*l* 38.6%

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

   6. *-commutative 38.6%

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

   7. distribute-rgt-neg-in 38.6%

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

   8. associate-*r/ 38.6%

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

   9. metadata-eval 38.6%

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

   10. distribute-neg-frac 38.6%

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

   11. metadata-eval 38.6%

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

5. Simplified 38.6%

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

6. Taylor expanded in z around 0 8.1%

   \[\leadsto \color{blue}{1} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024106 
(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)))