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

Percentage Accurate: 99.8% → 100.0%

Time: 6.2s

Alternatives: 7

Speedup: 1.4×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - 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 * 0.25d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))

C

double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - 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 * 0.25d0)) - z)) / y)
end function

Java

public static double code(double x, double y, double z) {
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
}

Python

def code(x, y, z):
	return 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y)

Julia

function code(x, y, z)
	return Float64(1.0 + Float64(Float64(4.0 * Float64(Float64(x + Float64(y * 0.25)) - z)) / y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = 1.0 + ((4.0 * ((x + (y * 0.25)) - z)) / y);
end

Wolfram

code[x_, y_, z_] := N[(1.0 + N[(N[(4.0 * N[(N[(x + N[(y * 0.25), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - z) / y)) + 2.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. associate-/l* 99.8%

      \[\leadsto 1 + \color{blue}{\frac{4}{\frac{y}{\left(x + y \cdot 0.25\right) - z}}} \]

   2. associate--l+ 99.8%

      \[\leadsto 1 + \frac{4}{\frac{y}{\color{blue}{x + \left(y \cdot 0.25 - z\right)}}} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{1 + \frac{4}{\frac{y}{x + \left(y \cdot 0.25 - z\right)}}} \]

4. Taylor expanded in y around 0 100.0%

   \[\leadsto \color{blue}{2 + 4 \cdot \frac{x - z}{y}} \]
5. Step-by-step derivation

   1. +-commutative 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+179}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+59} \lor \neg \left(y \leq 1.56 \cdot 10^{+94}\right) \land y \leq 4.8 \cdot 10^{+140}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.16e+179)
   2.0
   (if (or (<= y 9.2e+59) (and (not (<= y 1.56e+94)) (<= y 4.8e+140)))
     (* 4.0 (/ (- x z) y))
     2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.16e+179) {
		tmp = 2.0;
	} else if ((y <= 9.2e+59) || (!(y <= 1.56e+94) && (y <= 4.8e+140))) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.16d+179)) then
        tmp = 2.0d0
    else if ((y <= 9.2d+59) .or. (.not. (y <= 1.56d+94)) .and. (y <= 4.8d+140)) then
        tmp = 4.0d0 * ((x - z) / y)
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.16e+179) {
		tmp = 2.0;
	} else if ((y <= 9.2e+59) || (!(y <= 1.56e+94) && (y <= 4.8e+140))) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.16e+179:
		tmp = 2.0
	elif (y <= 9.2e+59) or (not (y <= 1.56e+94) and (y <= 4.8e+140)):
		tmp = 4.0 * ((x - z) / y)
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.16e+179)
		tmp = 2.0;
	elseif ((y <= 9.2e+59) || (!(y <= 1.56e+94) && (y <= 4.8e+140)))
		tmp = Float64(4.0 * Float64(Float64(x - z) / y));
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.16e+179)
		tmp = 2.0;
	elseif ((y <= 9.2e+59) || (~((y <= 1.56e+94)) && (y <= 4.8e+140)))
		tmp = 4.0 * ((x - z) / y);
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.16e+179], 2.0, If[Or[LessEqual[y, 9.2e+59], And[N[Not[LessEqual[y, 1.56e+94]], $MachinePrecision], LessEqual[y, 4.8e+140]]], N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], 2.0]]

Derivation

1. Split input into 2 regimes

2. if y < -1.16e179 or 9.20000000000000032e59 < y < 1.5600000000000001e94 or 4.7999999999999999e140 < y

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 100.0%

         \[\leadsto 1 + \color{blue}{\frac{4}{\frac{y}{\left(x + y \cdot 0.25\right) - z}}} \]

      2. associate--l+ 100.0%

         \[\leadsto 1 + \frac{4}{\frac{y}{\color{blue}{x + \left(y \cdot 0.25 - z\right)}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 + \frac{4}{\frac{y}{x + \left(y \cdot 0.25 - z\right)}}} \]

   4. Taylor expanded in y around inf 84.8%

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

   if -1.16e179 < y < 9.20000000000000032e59 or 1.5600000000000001e94 < y < 4.7999999999999999e140

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{\frac{y}{\left(x + y \cdot 0.25\right) - z}}} \]

      2. associate--l+ 99.7%

         \[\leadsto 1 + \frac{4}{\frac{y}{\color{blue}{x + \left(y \cdot 0.25 - z\right)}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{\frac{y}{x + \left(y \cdot 0.25 - z\right)}}} \]

   4. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{2 + 4 \cdot \frac{x - z}{y}} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 80.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.16 \cdot 10^{+179}:\\ \;\;\;\;2\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+59} \lor \neg \left(y \leq 1.56 \cdot 10^{+94}\right) \land y \leq 4.8 \cdot 10^{+140}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 3: 53.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ \mathbf{if}\;x \leq -9.5 \cdot 10^{-29}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 650:\\ \;\;\;\;2\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y))))
   (if (<= x -9.5e-29)
     t_0
     (if (<= x 650.0) 2.0 (if (<= x 3.8e+66) (* (/ z y) -4.0) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double tmp;
	if (x <= -9.5e-29) {
		tmp = t_0;
	} else if (x <= 650.0) {
		tmp = 2.0;
	} else if (x <= 3.8e+66) {
		tmp = (z / y) * -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 <= (-9.5d-29)) then
        tmp = t_0
    else if (x <= 650.0d0) then
        tmp = 2.0d0
    else if (x <= 3.8d+66) then
        tmp = (z / y) * (-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 <= -9.5e-29) {
		tmp = t_0;
	} else if (x <= 650.0) {
		tmp = 2.0;
	} else if (x <= 3.8e+66) {
		tmp = (z / y) * -4.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / y)
	tmp = 0
	if x <= -9.5e-29:
		tmp = t_0
	elif x <= 650.0:
		tmp = 2.0
	elif x <= 3.8e+66:
		tmp = (z / y) * -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 <= -9.5e-29)
		tmp = t_0;
	elseif (x <= 650.0)
		tmp = 2.0;
	elseif (x <= 3.8e+66)
		tmp = Float64(Float64(z / y) * -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 <= -9.5e-29)
		tmp = t_0;
	elseif (x <= 650.0)
		tmp = 2.0;
	elseif (x <= 3.8e+66)
		tmp = (z / y) * -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, -9.5e-29], t$95$0, If[LessEqual[x, 650.0], 2.0, If[LessEqual[x, 3.8e+66], N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if x < -9.50000000000000023e-29 or 3.8000000000000002e66 < x

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} \]

      2. +-commutative 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) \]

      3. associate--l+ 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} \]

      4. distribute-lft-in 99.7%

         \[\leadsto 1 + \color{blue}{\left(\frac{4}{y} \cdot \left(y \cdot 0.25\right) + \frac{4}{y} \cdot \left(x - z\right)\right)} \]

      5. associate-+r+ 99.7%

         \[\leadsto \color{blue}{\left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \frac{4}{y} \cdot \left(x - z\right)} \]

      6. *-commutative 99.7%

         \[\leadsto \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \color{blue}{\left(x - z\right) \cdot \frac{4}{y}} \]

      7. +-commutative 99.7%

         \[\leadsto \color{blue}{\left(x - z\right) \cdot \frac{4}{y} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} \]

      8. fma-def 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} \]

      9. *-commutative 99.7%

         \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{\left(y \cdot 0.25\right) \cdot \frac{4}{y}}\right) \]

      10. associate-*l* 99.7%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{y \cdot \left(0.25 \cdot \frac{4}{y}\right)}\right) \]

      11. associate-*r/ 99.7%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + y \cdot \color{blue}{\frac{0.25 \cdot 4}{y}}\right) \]

      12. metadata-eval 99.7%

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

      13. rgt-mult-inverse 99.7%

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

      14. metadata-eval 99.7%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, \color{blue}{2}\right) \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 2\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.7%

         \[\leadsto \color{blue}{\left(x - z\right) \cdot \frac{4}{y} + 2} \]

      2. flip-+ 47.4%

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

      3. metadata-eval 47.4%

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

   5. Applied egg-rr 47.4%

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

      1. associate-*r* 37.0%

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

      2. *-commutative 37.0%

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

      3. fma-neg 37.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{4}{y}, \left(\left(x - z\right) \cdot \frac{4}{y}\right) \cdot \left(x - z\right), -4\right)}}{\left(x - z\right) \cdot \frac{4}{y} - 2} \]

      4. *-commutative 37.0%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \color{blue}{\left(\frac{4}{y} \cdot \left(x - z\right)\right)} \cdot \left(x - z\right), -4\right)}{\left(x - z\right) \cdot \frac{4}{y} - 2} \]

      5. associate-*l* 23.9%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \color{blue}{\frac{4}{y} \cdot \left(\left(x - z\right) \cdot \left(x - z\right)\right)}, -4\right)}{\left(x - z\right) \cdot \frac{4}{y} - 2} \]

      6. unpow2 23.9%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot \color{blue}{{\left(x - z\right)}^{2}}, -4\right)}{\left(x - z\right) \cdot \frac{4}{y} - 2} \]

      7. metadata-eval 23.9%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, \color{blue}{-4}\right)}{\left(x - z\right) \cdot \frac{4}{y} - 2} \]

      8. *-commutative 23.9%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\color{blue}{\frac{4}{y} \cdot \left(x - z\right)} - 2} \]

      9. associate-*l/ 23.9%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\color{blue}{\frac{4 \cdot \left(x - z\right)}{y}} - 2} \]

      10. associate-*r/ 23.9%

          \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\color{blue}{4 \cdot \frac{x - z}{y}} - 2} \]

      11. fma-neg 23.9%

          \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\color{blue}{\mathsf{fma}\left(4, \frac{x - z}{y}, -2\right)}} \]

      12. metadata-eval 23.9%

          \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\mathsf{fma}\left(4, \frac{x - z}{y}, \color{blue}{-2}\right)} \]

   7. Simplified 23.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\mathsf{fma}\left(4, \frac{x - z}{y}, -2\right)}} \]

   8. Taylor expanded in z around inf 75.8%

      \[\leadsto \color{blue}{\left(2 + \left(8 \cdot \frac{x}{y} + -4 \cdot \frac{z}{y}\right)\right) - 4 \cdot \frac{x}{y}} \]

   9. Taylor expanded in x around inf 66.6%

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

   if -9.50000000000000023e-29 < x < 650

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{\frac{y}{\left(x + y \cdot 0.25\right) - z}}} \]

      2. associate--l+ 99.8%

         \[\leadsto 1 + \frac{4}{\frac{y}{\color{blue}{x + \left(y \cdot 0.25 - z\right)}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{4}{\frac{y}{x + \left(y \cdot 0.25 - z\right)}}} \]

   4. Taylor expanded in y around inf 56.1%

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

   if 650 < x < 3.8000000000000002e66

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.6%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} \]

      2. +-commutative 99.6%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) \]

      3. associate--l+ 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} \]

      4. distribute-lft-in 99.8%

         \[\leadsto 1 + \color{blue}{\left(\frac{4}{y} \cdot \left(y \cdot 0.25\right) + \frac{4}{y} \cdot \left(x - z\right)\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \frac{4}{y} \cdot \left(x - z\right)} \]

      6. *-commutative 99.8%

         \[\leadsto \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \color{blue}{\left(x - z\right) \cdot \frac{4}{y}} \]

      7. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(x - z\right) \cdot \frac{4}{y} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} \]

      8. fma-def 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} \]

      9. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{\left(y \cdot 0.25\right) \cdot \frac{4}{y}}\right) \]

      10. associate-*l* 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{y \cdot \left(0.25 \cdot \frac{4}{y}\right)}\right) \]

      11. associate-*r/ 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + y \cdot \color{blue}{\frac{0.25 \cdot 4}{y}}\right) \]

      12. metadata-eval 99.8%

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

      13. rgt-mult-inverse 99.8%

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

      14. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, \color{blue}{2}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 2\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.8%

         \[\leadsto \color{blue}{\left(x - z\right) \cdot \frac{4}{y} + 2} \]

      2. flip-+ 46.3%

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

      3. metadata-eval 46.3%

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

   5. Applied egg-rr 46.3%

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

      1. associate-*r* 46.3%

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

      2. *-commutative 46.3%

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

      3. fma-neg 46.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{4}{y}, \left(\left(x - z\right) \cdot \frac{4}{y}\right) \cdot \left(x - z\right), -4\right)}}{\left(x - z\right) \cdot \frac{4}{y} - 2} \]

      4. *-commutative 46.3%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \color{blue}{\left(\frac{4}{y} \cdot \left(x - z\right)\right)} \cdot \left(x - z\right), -4\right)}{\left(x - z\right) \cdot \frac{4}{y} - 2} \]

      5. associate-*l* 38.7%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \color{blue}{\frac{4}{y} \cdot \left(\left(x - z\right) \cdot \left(x - z\right)\right)}, -4\right)}{\left(x - z\right) \cdot \frac{4}{y} - 2} \]

      6. unpow2 38.7%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot \color{blue}{{\left(x - z\right)}^{2}}, -4\right)}{\left(x - z\right) \cdot \frac{4}{y} - 2} \]

      7. metadata-eval 38.7%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, \color{blue}{-4}\right)}{\left(x - z\right) \cdot \frac{4}{y} - 2} \]

      8. *-commutative 38.7%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\color{blue}{\frac{4}{y} \cdot \left(x - z\right)} - 2} \]

      9. associate-*l/ 38.5%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\color{blue}{\frac{4 \cdot \left(x - z\right)}{y}} - 2} \]

      10. associate-*r/ 38.5%

          \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\color{blue}{4 \cdot \frac{x - z}{y}} - 2} \]

      11. fma-neg 38.5%

          \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\color{blue}{\mathsf{fma}\left(4, \frac{x - z}{y}, -2\right)}} \]

      12. metadata-eval 38.5%

          \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\mathsf{fma}\left(4, \frac{x - z}{y}, \color{blue}{-2}\right)} \]

   7. Simplified 38.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\mathsf{fma}\left(4, \frac{x - z}{y}, -2\right)}} \]

   8. Taylor expanded in z around inf 99.8%

      \[\leadsto \color{blue}{\left(2 + \left(8 \cdot \frac{x}{y} + -4 \cdot \frac{z}{y}\right)\right) - 4 \cdot \frac{x}{y}} \]

   9. Taylor expanded in z around inf 69.0%

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

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-29}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq 650:\\ \;\;\;\;2\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+66}:\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 4: 85.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+115} \lor \neg \left(z \leq 1.7 \cdot 10^{+80}\right):\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.85e+115) (not (<= z 1.7e+80)))
   (* 4.0 (/ (- x z) y))
   (+ 2.0 (* 4.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.85e+115) || !(z <= 1.7e+80)) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 2.0 + (4.0 * (x / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.85d+115)) .or. (.not. (z <= 1.7d+80))) then
        tmp = 4.0d0 * ((x - z) / y)
    else
        tmp = 2.0d0 + (4.0d0 * (x / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.85e+115) || !(z <= 1.7e+80)) {
		tmp = 4.0 * ((x - z) / y);
	} else {
		tmp = 2.0 + (4.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.85e+115) or not (z <= 1.7e+80):
		tmp = 4.0 * ((x - z) / y)
	else:
		tmp = 2.0 + (4.0 * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.85e+115) || !(z <= 1.7e+80))
		tmp = Float64(4.0 * Float64(Float64(x - z) / y));
	else
		tmp = Float64(2.0 + Float64(4.0 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.85e+115) || ~((z <= 1.7e+80)))
		tmp = 4.0 * ((x - z) / y);
	else
		tmp = 2.0 + (4.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.85e+115], N[Not[LessEqual[z, 1.7e+80]], $MachinePrecision]], N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.85000000000000003e115 or 1.69999999999999996e80 < z

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{\frac{y}{\left(x + y \cdot 0.25\right) - z}}} \]

      2. associate--l+ 99.8%

         \[\leadsto 1 + \frac{4}{\frac{y}{\color{blue}{x + \left(y \cdot 0.25 - z\right)}}} \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{1 + \frac{4}{\frac{y}{x + \left(y \cdot 0.25 - z\right)}}} \]

   4. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{2 + 4 \cdot \frac{x - z}{y}} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 89.7%

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

   if -1.85000000000000003e115 < z < 1.69999999999999996e80

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} \]

      2. +-commutative 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) \]

      3. associate--l+ 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} \]

      4. distribute-lft-in 99.8%

         \[\leadsto 1 + \color{blue}{\left(\frac{4}{y} \cdot \left(y \cdot 0.25\right) + \frac{4}{y} \cdot \left(x - z\right)\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \frac{4}{y} \cdot \left(x - z\right)} \]

      6. *-commutative 99.8%

         \[\leadsto \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \color{blue}{\left(x - z\right) \cdot \frac{4}{y}} \]

      7. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(x - z\right) \cdot \frac{4}{y} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} \]

      8. fma-def 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} \]

      9. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{\left(y \cdot 0.25\right) \cdot \frac{4}{y}}\right) \]

      10. associate-*l* 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{y \cdot \left(0.25 \cdot \frac{4}{y}\right)}\right) \]

      11. associate-*r/ 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + y \cdot \color{blue}{\frac{0.25 \cdot 4}{y}}\right) \]

      12. metadata-eval 99.8%

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

      13. rgt-mult-inverse 99.8%

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

      14. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, \color{blue}{2}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 2\right)} \]

   4. Taylor expanded in z around 0 90.6%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+115} \lor \neg \left(z \leq 1.7 \cdot 10^{+80}\right):\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \]

Alternative 5: 86.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+44}:\\ \;\;\;\;2 + \frac{z}{y} \cdot -4\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+78}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z -2.6e+44)
   (+ 2.0 (* (/ z y) -4.0))
   (if (<= z 1.9e+78) (+ 2.0 (* 4.0 (/ x y))) (* 4.0 (/ (- x z) y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= -2.6e+44) {
		tmp = 2.0 + ((z / y) * -4.0);
	} else if (z <= 1.9e+78) {
		tmp = 2.0 + (4.0 * (x / y));
	} else {
		tmp = 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 (z <= (-2.6d+44)) then
        tmp = 2.0d0 + ((z / y) * (-4.0d0))
    else if (z <= 1.9d+78) then
        tmp = 2.0d0 + (4.0d0 * (x / y))
    else
        tmp = 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 (z <= -2.6e+44) {
		tmp = 2.0 + ((z / y) * -4.0);
	} else if (z <= 1.9e+78) {
		tmp = 2.0 + (4.0 * (x / y));
	} else {
		tmp = 4.0 * ((x - z) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= -2.6e+44:
		tmp = 2.0 + ((z / y) * -4.0)
	elif z <= 1.9e+78:
		tmp = 2.0 + (4.0 * (x / y))
	else:
		tmp = 4.0 * ((x - z) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= -2.6e+44)
		tmp = Float64(2.0 + Float64(Float64(z / y) * -4.0));
	elseif (z <= 1.9e+78)
		tmp = Float64(2.0 + Float64(4.0 * Float64(x / y)));
	else
		tmp = Float64(4.0 * Float64(Float64(x - z) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= -2.6e+44)
		tmp = 2.0 + ((z / y) * -4.0);
	elseif (z <= 1.9e+78)
		tmp = 2.0 + (4.0 * (x / y));
	else
		tmp = 4.0 * ((x - z) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, -2.6e+44], N[(2.0 + N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.9e+78], N[(2.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.5999999999999999e44

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) \]

      3. associate--l+ 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} \]

      4. distribute-lft-in 99.8%

         \[\leadsto 1 + \color{blue}{\left(\frac{4}{y} \cdot \left(y \cdot 0.25\right) + \frac{4}{y} \cdot \left(x - z\right)\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \frac{4}{y} \cdot \left(x - z\right)} \]

      6. *-commutative 99.8%

         \[\leadsto \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \color{blue}{\left(x - z\right) \cdot \frac{4}{y}} \]

      7. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(x - z\right) \cdot \frac{4}{y} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} \]

      8. fma-def 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} \]

      9. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{\left(y \cdot 0.25\right) \cdot \frac{4}{y}}\right) \]

      10. associate-*l* 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{y \cdot \left(0.25 \cdot \frac{4}{y}\right)}\right) \]

      11. associate-*r/ 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + y \cdot \color{blue}{\frac{0.25 \cdot 4}{y}}\right) \]

      12. metadata-eval 99.8%

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

      13. rgt-mult-inverse 99.8%

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

      14. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, \color{blue}{2}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 2\right)} \]

   4. Taylor expanded in x around 0 88.0%

      \[\leadsto \color{blue}{2 + -4 \cdot \frac{z}{y}} \]
   5. Step-by-step derivation

      1. +-commutative 88.0%

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

      2. *-commutative 88.0%

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

   6. Simplified 88.0%

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

   if -2.5999999999999999e44 < z < 1.9e78

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} \]

      2. +-commutative 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) \]

      3. associate--l+ 99.7%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} \]

      4. distribute-lft-in 99.7%

         \[\leadsto 1 + \color{blue}{\left(\frac{4}{y} \cdot \left(y \cdot 0.25\right) + \frac{4}{y} \cdot \left(x - z\right)\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \frac{4}{y} \cdot \left(x - z\right)} \]

      6. *-commutative 99.8%

         \[\leadsto \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \color{blue}{\left(x - z\right) \cdot \frac{4}{y}} \]

      7. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(x - z\right) \cdot \frac{4}{y} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} \]

      8. fma-def 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} \]

      9. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{\left(y \cdot 0.25\right) \cdot \frac{4}{y}}\right) \]

      10. associate-*l* 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{y \cdot \left(0.25 \cdot \frac{4}{y}\right)}\right) \]

      11. associate-*r/ 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + y \cdot \color{blue}{\frac{0.25 \cdot 4}{y}}\right) \]

      12. metadata-eval 99.8%

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

      13. rgt-mult-inverse 99.8%

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

      14. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, \color{blue}{2}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 2\right)} \]

   4. Taylor expanded in z around 0 92.3%

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

   if 1.9e78 < z

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{\frac{y}{\left(x + y \cdot 0.25\right) - z}}} \]

      2. associate--l+ 99.7%

         \[\leadsto 1 + \frac{4}{\frac{y}{\color{blue}{x + \left(y \cdot 0.25 - z\right)}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{\frac{y}{x + \left(y \cdot 0.25 - z\right)}}} \]

   4. Taylor expanded in y around 0 100.0%

      \[\leadsto \color{blue}{2 + 4 \cdot \frac{x - z}{y}} \]
   5. Step-by-step derivation

      1. +-commutative 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 90.6%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+44}:\\ \;\;\;\;2 + \frac{z}{y} \cdot -4\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+78}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x - z}{y}\\ \end{array} \]

Alternative 6: 52.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.82 \cdot 10^{+115} \lor \neg \left(z \leq 1.55 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.82e+115) (not (<= z 1.55e+100))) (* (/ z y) -4.0) 2.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.82e+115) || !(z <= 1.55e+100)) {
		tmp = (z / y) * -4.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.82d+115)) .or. (.not. (z <= 1.55d+100))) then
        tmp = (z / y) * (-4.0d0)
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.82e+115) || !(z <= 1.55e+100)) {
		tmp = (z / y) * -4.0;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.82e+115) or not (z <= 1.55e+100):
		tmp = (z / y) * -4.0
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.82e+115) || !(z <= 1.55e+100))
		tmp = Float64(Float64(z / y) * -4.0);
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.82e+115) || ~((z <= 1.55e+100)))
		tmp = (z / y) * -4.0;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.82e+115], N[Not[LessEqual[z, 1.55e+100]], $MachinePrecision]], N[(N[(z / y), $MachinePrecision] * -4.0), $MachinePrecision], 2.0]

Derivation

1. Split input into 2 regimes

2. if z < -1.82e115 or 1.55000000000000003e100 < z

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-*l/ 99.8%

         \[\leadsto 1 + \color{blue}{\frac{4}{y} \cdot \left(\left(x + y \cdot 0.25\right) - z\right)} \]

      2. +-commutative 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \left(\color{blue}{\left(y \cdot 0.25 + x\right)} - z\right) \]

      3. associate--l+ 99.8%

         \[\leadsto 1 + \frac{4}{y} \cdot \color{blue}{\left(y \cdot 0.25 + \left(x - z\right)\right)} \]

      4. distribute-lft-in 99.8%

         \[\leadsto 1 + \color{blue}{\left(\frac{4}{y} \cdot \left(y \cdot 0.25\right) + \frac{4}{y} \cdot \left(x - z\right)\right)} \]

      5. associate-+r+ 99.8%

         \[\leadsto \color{blue}{\left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \frac{4}{y} \cdot \left(x - z\right)} \]

      6. *-commutative 99.8%

         \[\leadsto \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right) + \color{blue}{\left(x - z\right) \cdot \frac{4}{y}} \]

      7. +-commutative 99.8%

         \[\leadsto \color{blue}{\left(x - z\right) \cdot \frac{4}{y} + \left(1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} \]

      8. fma-def 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \frac{4}{y} \cdot \left(y \cdot 0.25\right)\right)} \]

      9. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{\left(y \cdot 0.25\right) \cdot \frac{4}{y}}\right) \]

      10. associate-*l* 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + \color{blue}{y \cdot \left(0.25 \cdot \frac{4}{y}\right)}\right) \]

      11. associate-*r/ 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, 1 + y \cdot \color{blue}{\frac{0.25 \cdot 4}{y}}\right) \]

      12. metadata-eval 99.8%

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

      13. rgt-mult-inverse 99.8%

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

      14. metadata-eval 99.8%

          \[\leadsto \mathsf{fma}\left(x - z, \frac{4}{y}, \color{blue}{2}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - z, \frac{4}{y}, 2\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.8%

         \[\leadsto \color{blue}{\left(x - z\right) \cdot \frac{4}{y} + 2} \]

      2. flip-+ 30.5%

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

      3. metadata-eval 30.5%

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

   5. Applied egg-rr 30.5%

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

      1. associate-*r* 23.1%

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

      2. *-commutative 23.1%

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

      3. fma-neg 23.1%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{4}{y}, \left(\left(x - z\right) \cdot \frac{4}{y}\right) \cdot \left(x - z\right), -4\right)}}{\left(x - z\right) \cdot \frac{4}{y} - 2} \]

      4. *-commutative 23.1%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \color{blue}{\left(\frac{4}{y} \cdot \left(x - z\right)\right)} \cdot \left(x - z\right), -4\right)}{\left(x - z\right) \cdot \frac{4}{y} - 2} \]

      5. associate-*l* 9.2%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \color{blue}{\frac{4}{y} \cdot \left(\left(x - z\right) \cdot \left(x - z\right)\right)}, -4\right)}{\left(x - z\right) \cdot \frac{4}{y} - 2} \]

      6. unpow2 9.2%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot \color{blue}{{\left(x - z\right)}^{2}}, -4\right)}{\left(x - z\right) \cdot \frac{4}{y} - 2} \]

      7. metadata-eval 9.2%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, \color{blue}{-4}\right)}{\left(x - z\right) \cdot \frac{4}{y} - 2} \]

      8. *-commutative 9.2%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\color{blue}{\frac{4}{y} \cdot \left(x - z\right)} - 2} \]

      9. associate-*l/ 9.2%

         \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\color{blue}{\frac{4 \cdot \left(x - z\right)}{y}} - 2} \]

      10. associate-*r/ 9.2%

          \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\color{blue}{4 \cdot \frac{x - z}{y}} - 2} \]

      11. fma-neg 9.2%

          \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\color{blue}{\mathsf{fma}\left(4, \frac{x - z}{y}, -2\right)}} \]

      12. metadata-eval 9.2%

          \[\leadsto \frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\mathsf{fma}\left(4, \frac{x - z}{y}, \color{blue}{-2}\right)} \]

   7. Simplified 9.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{4}{y}, \frac{4}{y} \cdot {\left(x - z\right)}^{2}, -4\right)}{\mathsf{fma}\left(4, \frac{x - z}{y}, -2\right)}} \]

   8. Taylor expanded in z around inf 84.4%

      \[\leadsto \color{blue}{\left(2 + \left(8 \cdot \frac{x}{y} + -4 \cdot \frac{z}{y}\right)\right) - 4 \cdot \frac{x}{y}} \]

   9. Taylor expanded in z around inf 79.3%

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

   if -1.82e115 < z < 1.55000000000000003e100

   1. Initial program 99.9%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Step-by-step derivation

      1. associate-/l* 99.7%

         \[\leadsto 1 + \color{blue}{\frac{4}{\frac{y}{\left(x + y \cdot 0.25\right) - z}}} \]

      2. associate--l+ 99.7%

         \[\leadsto 1 + \frac{4}{\frac{y}{\color{blue}{x + \left(y \cdot 0.25 - z\right)}}} \]

   3. Simplified 99.7%

      \[\leadsto \color{blue}{1 + \frac{4}{\frac{y}{x + \left(y \cdot 0.25 - z\right)}}} \]

   4. Taylor expanded in y around inf 46.4%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.82 \cdot 10^{+115} \lor \neg \left(z \leq 1.55 \cdot 10^{+100}\right):\\ \;\;\;\;\frac{z}{y} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]

Alternative 7: 33.7% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 2.0)

C

double code(double x, double y, double z) {
	return 2.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return 2.0;
}

Python

def code(x, y, z):
	return 2.0

Julia

function code(x, y, z)
	return 2.0
end

MATLAB

function tmp = code(x, y, z)
	tmp = 2.0;
end

Wolfram

code[x_, y_, z_] := 2.0

Derivation

1. Initial program 100.0%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
2. Step-by-step derivation

   1. associate-/l* 99.8%

      \[\leadsto 1 + \color{blue}{\frac{4}{\frac{y}{\left(x + y \cdot 0.25\right) - z}}} \]

   2. associate--l+ 99.8%

      \[\leadsto 1 + \frac{4}{\frac{y}{\color{blue}{x + \left(y \cdot 0.25 - z\right)}}} \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{1 + \frac{4}{\frac{y}{x + \left(y \cdot 0.25 - z\right)}}} \]

4. Taylor expanded in y around inf 35.7%

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

5. Final simplification 35.7%

   \[\leadsto 2 \]

Reproduce

herbie shell --seed 2023185 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, C"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.25)) z)) y)))