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

Percentage Accurate: 99.8% → 100.0%

Time: 5.9s

Alternatives: 9

Speedup: 1.4×

• 21.3% 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: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

3. Taylor expanded in y around inf 100.0%

   \[\leadsto \color{blue}{4 + 4 \cdot \frac{x - z}{y}} \]
4. Add Preprocessing

Alternative 2: 52.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -3.5 \cdot 10^{+64}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-102}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+105}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y))))
   (if (<= x -3.5e+64)
     t_0
     (if (<= x 1.12e-102) (* -4.0 (/ z y)) (if (<= x 3.8e+105) 4.0 t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double tmp;
	if (x <= -3.5e+64) {
		tmp = t_0;
	} else if (x <= 1.12e-102) {
		tmp = -4.0 * (z / y);
	} else if (x <= 3.8e+105) {
		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 = 4.0d0 * (x / y)
    if (x <= (-3.5d+64)) then
        tmp = t_0
    else if (x <= 1.12d-102) then
        tmp = (-4.0d0) * (z / y)
    else if (x <= 3.8d+105) 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 = 4.0 * (x / y);
	double tmp;
	if (x <= -3.5e+64) {
		tmp = t_0;
	} else if (x <= 1.12e-102) {
		tmp = -4.0 * (z / y);
	} else if (x <= 3.8e+105) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / y)
	tmp = 0
	if x <= -3.5e+64:
		tmp = t_0
	elif x <= 1.12e-102:
		tmp = -4.0 * (z / y)
	elif x <= 3.8e+105:
		tmp = 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / y))
	tmp = 0.0
	if (x <= -3.5e+64)
		tmp = t_0;
	elseif (x <= 1.12e-102)
		tmp = Float64(-4.0 * Float64(z / y));
	elseif (x <= 3.8e+105)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / y);
	tmp = 0.0;
	if (x <= -3.5e+64)
		tmp = t_0;
	elseif (x <= 1.12e-102)
		tmp = -4.0 * (z / y);
	elseif (x <= 3.8e+105)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.5e+64], t$95$0, If[LessEqual[x, 1.12e-102], N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.8e+105], 4.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.4999999999999999e64 or 3.8e105 < x

   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 inf 72.9%

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

   if -3.4999999999999999e64 < x < 1.12000000000000009e-102

   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 51.0%

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

      1. *-commutative 51.0%

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

   5. Simplified 51.0%

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

   if 1.12000000000000009e-102 < x < 3.8e105

   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 66.5%

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

4. Final simplification 61.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.5 \cdot 10^{+64}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-102}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+105}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{+65}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-101}:\\ \;\;\;\;z \cdot \frac{-4}{y}\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{+105}:\\ \;\;\;\;4\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y))))
   (if (<= x -3.2e+65)
     t_0
     (if (<= x 1.1e-101) (* z (/ -4.0 y)) (if (<= x 1.12e+105) 4.0 t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double tmp;
	if (x <= -3.2e+65) {
		tmp = t_0;
	} else if (x <= 1.1e-101) {
		tmp = z * (-4.0 / y);
	} else if (x <= 1.12e+105) {
		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 = 4.0d0 * (x / y)
    if (x <= (-3.2d+65)) then
        tmp = t_0
    else if (x <= 1.1d-101) then
        tmp = z * ((-4.0d0) / y)
    else if (x <= 1.12d+105) 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 = 4.0 * (x / y);
	double tmp;
	if (x <= -3.2e+65) {
		tmp = t_0;
	} else if (x <= 1.1e-101) {
		tmp = z * (-4.0 / y);
	} else if (x <= 1.12e+105) {
		tmp = 4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / y)
	tmp = 0
	if x <= -3.2e+65:
		tmp = t_0
	elif x <= 1.1e-101:
		tmp = z * (-4.0 / y)
	elif x <= 1.12e+105:
		tmp = 4.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / y))
	tmp = 0.0
	if (x <= -3.2e+65)
		tmp = t_0;
	elseif (x <= 1.1e-101)
		tmp = Float64(z * Float64(-4.0 / y));
	elseif (x <= 1.12e+105)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / y);
	tmp = 0.0;
	if (x <= -3.2e+65)
		tmp = t_0;
	elseif (x <= 1.1e-101)
		tmp = z * (-4.0 / y);
	elseif (x <= 1.12e+105)
		tmp = 4.0;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.2e+65], t$95$0, If[LessEqual[x, 1.1e-101], N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.12e+105], 4.0, t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.20000000000000007e65 or 1.12e105 < x

   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 inf 72.9%

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

   if -3.20000000000000007e65 < x < 1.0999999999999999e-101

   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 51.0%

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

      1. associate-*r/ 51.0%

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

      2. *-commutative 51.0%

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

      3. associate-/l* 50.8%

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

   5. Simplified 50.8%

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

   if 1.0999999999999999e-101 < x < 1.12e105

   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 66.5%

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

Alternative 4: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+84} \lor \neg \left(y \leq 2.9 \cdot 10^{+88}\right):\\ \;\;\;\;4 + x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.4e+84) (not (<= y 2.9e+88)))
   (+ 4.0 (* x (/ 4.0 y)))
   (* (- x z) (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.4e+84) || !(y <= 2.9e+88)) {
		tmp = 4.0 + (x * (4.0 / y));
	} else {
		tmp = (x - 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 ((y <= (-1.4d+84)) .or. (.not. (y <= 2.9d+88))) then
        tmp = 4.0d0 + (x * (4.0d0 / y))
    else
        tmp = (x - z) * (4.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.4e+84) || !(y <= 2.9e+88)) {
		tmp = 4.0 + (x * (4.0 / y));
	} else {
		tmp = (x - z) * (4.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -1.4e+84) or not (y <= 2.9e+88):
		tmp = 4.0 + (x * (4.0 / y))
	else:
		tmp = (x - z) * (4.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.4e+84) || !(y <= 2.9e+88))
		tmp = Float64(4.0 + Float64(x * Float64(4.0 / y)));
	else
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -1.4e+84) || ~((y <= 2.9e+88)))
		tmp = 4.0 + (x * (4.0 / y));
	else
		tmp = (x - z) * (4.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.4e+84], N[Not[LessEqual[y, 2.9e+88]], $MachinePrecision]], N[(4.0 + N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.39999999999999991e84 or 2.9e88 < y

   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. +-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. associate--r+ 99.9%

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

      9. div-sub 99.9%

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

      10. sub-neg 99.9%

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

      11. associate-*l/ 100.0%

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

      12. *-inverses 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-frac-neg2 100.0%

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

      15. remove-double-neg 100.0%

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

      16. distribute-neg-out 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. distribute-frac-neg2 100.0%

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

      21. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 87.1%

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

      1. distribute-lft-in 87.1%

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

      2. metadata-eval 87.1%

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

      3. associate-+r+ 87.1%

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

      4. metadata-eval 87.1%

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

      5. associate-*r/ 87.1%

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

      6. *-commutative 87.1%

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

      7. associate-*r/ 87.0%

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

   7. Simplified 87.0%

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

   if -1.39999999999999991e84 < y < 2.9e88

   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 0 87.9%

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

      1. *-lft-identity 87.9%

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

      2. associate-*l/ 87.6%

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

      3. associate-*r* 87.6%

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

      4. associate-*r/ 87.6%

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

      5. metadata-eval 87.6%

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

   5. Simplified 87.6%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+84} \lor \neg \left(y \leq 2.9 \cdot 10^{+88}\right):\\ \;\;\;\;4 + x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+20}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+37}:\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + x \cdot \frac{4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -9e+20)
   (* (- x z) (/ 4.0 y))
   (if (<= x 6.2e+37) (+ 4.0 (/ (* z -4.0) y)) (+ 4.0 (* x (/ 4.0 y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -9e+20) {
		tmp = (x - z) * (4.0 / y);
	} else if (x <= 6.2e+37) {
		tmp = 4.0 + ((z * -4.0) / y);
	} else {
		tmp = 4.0 + (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 <= (-9d+20)) then
        tmp = (x - z) * (4.0d0 / y)
    else if (x <= 6.2d+37) then
        tmp = 4.0d0 + ((z * (-4.0d0)) / y)
    else
        tmp = 4.0d0 + (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 <= -9e+20) {
		tmp = (x - z) * (4.0 / y);
	} else if (x <= 6.2e+37) {
		tmp = 4.0 + ((z * -4.0) / y);
	} else {
		tmp = 4.0 + (x * (4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -9e+20:
		tmp = (x - z) * (4.0 / y)
	elif x <= 6.2e+37:
		tmp = 4.0 + ((z * -4.0) / y)
	else:
		tmp = 4.0 + (x * (4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -9e+20)
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	elseif (x <= 6.2e+37)
		tmp = Float64(4.0 + Float64(Float64(z * -4.0) / y));
	else
		tmp = Float64(4.0 + Float64(x * Float64(4.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -9e+20)
		tmp = (x - z) * (4.0 / y);
	elseif (x <= 6.2e+37)
		tmp = 4.0 + ((z * -4.0) / y);
	else
		tmp = 4.0 + (x * (4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -9e+20], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.2e+37], N[(4.0 + N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -9e20

   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 0 83.3%

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

      1. *-lft-identity 83.3%

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

      2. associate-*l/ 83.0%

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

      3. associate-*r* 83.0%

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

      4. associate-*r/ 83.0%

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

      5. metadata-eval 83.0%

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

   5. Simplified 83.0%

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

   if -9e20 < x < 6.2000000000000004e37

   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%

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

   4. Taylor expanded in x around 0 92.4%

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

      1. +-commutative 92.4%

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

      2. associate-*r/ 92.4%

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

   6. Simplified 92.4%

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

   if 6.2000000000000004e37 < 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. +-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. associate--r+ 99.9%

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

      9. div-sub 99.9%

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

      10. sub-neg 99.9%

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

      11. associate-*l/ 100.0%

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

      12. *-inverses 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-frac-neg2 100.0%

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

      15. remove-double-neg 100.0%

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

      16. distribute-neg-out 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. distribute-frac-neg2 100.0%

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

      21. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 93.4%

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

      1. distribute-lft-in 93.4%

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

      2. metadata-eval 93.4%

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

      3. associate-+r+ 93.4%

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

      4. metadata-eval 93.4%

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

      5. associate-*r/ 93.4%

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

      6. *-commutative 93.4%

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

      7. associate-*r/ 93.2%

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

   7. Simplified 93.2%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{+20}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+37}:\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + x \cdot \frac{4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+21}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+36}:\\ \;\;\;\;4 + z \cdot \frac{-4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + x \cdot \frac{4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -6.2e+21)
   (* (- x z) (/ 4.0 y))
   (if (<= x 3.7e+36) (+ 4.0 (* z (/ -4.0 y))) (+ 4.0 (* x (/ 4.0 y))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -6.2e+21) {
		tmp = (x - z) * (4.0 / y);
	} else if (x <= 3.7e+36) {
		tmp = 4.0 + (z * (-4.0 / y));
	} else {
		tmp = 4.0 + (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 <= (-6.2d+21)) then
        tmp = (x - z) * (4.0d0 / y)
    else if (x <= 3.7d+36) then
        tmp = 4.0d0 + (z * ((-4.0d0) / y))
    else
        tmp = 4.0d0 + (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 <= -6.2e+21) {
		tmp = (x - z) * (4.0 / y);
	} else if (x <= 3.7e+36) {
		tmp = 4.0 + (z * (-4.0 / y));
	} else {
		tmp = 4.0 + (x * (4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -6.2e+21:
		tmp = (x - z) * (4.0 / y)
	elif x <= 3.7e+36:
		tmp = 4.0 + (z * (-4.0 / y))
	else:
		tmp = 4.0 + (x * (4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -6.2e+21)
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	elseif (x <= 3.7e+36)
		tmp = Float64(4.0 + Float64(z * Float64(-4.0 / y)));
	else
		tmp = Float64(4.0 + Float64(x * Float64(4.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -6.2e+21)
		tmp = (x - z) * (4.0 / y);
	elseif (x <= 3.7e+36)
		tmp = 4.0 + (z * (-4.0 / y));
	else
		tmp = 4.0 + (x * (4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -6.2e+21], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 3.7e+36], N[(4.0 + N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.2e21

   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 0 83.3%

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

      1. *-lft-identity 83.3%

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

      2. associate-*l/ 83.0%

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

      3. associate-*r* 83.0%

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

      4. associate-*r/ 83.0%

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

      5. metadata-eval 83.0%

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

   5. Simplified 83.0%

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

   if -6.2e21 < x < 3.70000000000000029e36

   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. +-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. associate--r+ 99.9%

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

      9. div-sub 100.0%

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

      10. sub-neg 100.0%

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

      11. associate-*l/ 100.0%

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

      12. *-inverses 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-frac-neg2 100.0%

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

      15. remove-double-neg 100.0%

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

      16. distribute-neg-out 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. distribute-frac-neg2 100.0%

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

      21. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 92.4%

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

      1. sub-neg 92.4%

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

      2. distribute-lft-in 92.4%

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

      3. metadata-eval 92.4%

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

      4. associate-+r+ 92.4%

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

      5. metadata-eval 92.4%

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

      6. neg-mul-1 92.4%

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

      7. associate-*r* 92.4%

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

      8. metadata-eval 92.4%

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

      9. associate-*r/ 92.4%

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

      10. *-commutative 92.4%

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

      11. associate-/l* 92.2%

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

   7. Simplified 92.2%

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

   if 3.70000000000000029e36 < 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. +-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. associate--r+ 99.9%

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

      9. div-sub 99.9%

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

      10. sub-neg 99.9%

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

      11. associate-*l/ 100.0%

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

      12. *-inverses 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-frac-neg2 100.0%

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

      15. remove-double-neg 100.0%

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

      16. distribute-neg-out 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. distribute-frac-neg2 100.0%

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

      21. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 93.4%

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

      1. distribute-lft-in 93.4%

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

      2. metadata-eval 93.4%

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

      3. associate-+r+ 93.4%

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

      4. metadata-eval 93.4%

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

      5. associate-*r/ 93.4%

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

      6. *-commutative 93.4%

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

      7. associate-*r/ 93.2%

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

   7. Simplified 93.2%

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

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.2 \cdot 10^{+21}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{+36}:\\ \;\;\;\;4 + z \cdot \frac{-4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + x \cdot \frac{4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+145}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+99}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.5e+145) 4.0 (if (<= y 1.35e+99) (* (- x z) (/ 4.0 y)) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -2.5e+145) {
		tmp = 4.0;
	} else if (y <= 1.35e+99) {
		tmp = (x - z) * (4.0 / 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 <= (-2.5d+145)) then
        tmp = 4.0d0
    else if (y <= 1.35d+99) then
        tmp = (x - z) * (4.0d0 / 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 <= -2.5e+145) {
		tmp = 4.0;
	} else if (y <= 1.35e+99) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -2.5e+145:
		tmp = 4.0
	elif y <= 1.35e+99:
		tmp = (x - z) * (4.0 / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -2.5e+145)
		tmp = 4.0;
	elseif (y <= 1.35e+99)
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -2.5e+145)
		tmp = 4.0;
	elseif (y <= 1.35e+99)
		tmp = (x - z) * (4.0 / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -2.5e+145], 4.0, If[LessEqual[y, 1.35e+99], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -2.49999999999999983e145 or 1.34999999999999994e99 < 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 75.9%

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

   if -2.49999999999999983e145 < y < 1.34999999999999994e99

   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 0 85.3%

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

      1. *-lft-identity 85.3%

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

      2. associate-*l/ 85.0%

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

      3. associate-*r* 85.0%

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

      4. associate-*r/ 85.0%

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

      5. metadata-eval 85.0%

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

   5. Simplified 85.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{+145}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 1.35 \cdot 10^{+99}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 52.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+27} \lor \neg \left(x \leq 1.11 \cdot 10^{+105}\right):\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.65e+27) (not (<= x 1.11e+105))) (* 4.0 (/ x y)) 4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.65e+27) || !(x <= 1.11e+105)) {
		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 ((x <= (-1.65d+27)) .or. (.not. (x <= 1.11d+105))) 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 ((x <= -1.65e+27) || !(x <= 1.11e+105)) {
		tmp = 4.0 * (x / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.65e+27) or not (x <= 1.11e+105):
		tmp = 4.0 * (x / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.65e+27) || !(x <= 1.11e+105))
		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 ((x <= -1.65e+27) || ~((x <= 1.11e+105)))
		tmp = 4.0 * (x / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.65e+27], N[Not[LessEqual[x, 1.11e+105]], $MachinePrecision]], N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.6499999999999999e27 or 1.11e105 < x

   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 inf 69.5%

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

   if -1.6499999999999999e27 < x < 1.11e105

   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 48.5%

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

4. Final simplification 57.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+27} \lor \neg \left(x \leq 1.11 \cdot 10^{+105}\right):\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.2% 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 34.8%

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

Reproduce

herbie shell --seed 2024165 
(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)))