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

Percentage Accurate: 99.9% → 100.0%

Time: 5.6s

Alternatives: 10

Speedup: 1.4×

• 20.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

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: 99.9% 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

\[\begin{array}{l} \\ \frac{x - z}{y \cdot 0.25} + 2 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (/ (- x z) (* y 0.25)) 2.0))

C

double code(double x, double y, double z) {
	return ((x - z) / (y * 0.25)) + 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 = ((x - z) / (y * 0.25d0)) + 2.0d0
end function

Java

public static double code(double x, double y, double z) {
	return ((x - z) / (y * 0.25)) + 2.0;
}

Python

def code(x, y, z):
	return ((x - z) / (y * 0.25)) + 2.0

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x - z) / Float64(y * 0.25)) + 2.0)
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x - z) / (y * 0.25)) + 2.0;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x - z), $MachinePrecision] / N[(y * 0.25), $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. +-commutative 100.0%

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

   2. associate-*l/ 99.8%

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

   3. +-commutative 99.8%

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

   4. associate--l+ 99.8%

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

   5. +-commutative 99.8%

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

   6. distribute-lft-in 99.4%

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

   7. associate-+l+ 99.4%

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

   8. associate-*l/ 99.8%

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

   9. *-commutative 99.8%

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

   10. associate-*l* 99.8%

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

   11. metadata-eval 99.8%

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

   12. *-rgt-identity 99.8%

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

   13. *-inverses 99.8%

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

   14. metadata-eval 99.8%

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

3. Simplified 99.8%

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

   1. clear-num 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{4}}} \cdot \left(x - z\right) + 2 \]

   2. div-inv 99.8%

      \[\leadsto \frac{1}{\color{blue}{y \cdot \frac{1}{4}}} \cdot \left(x - z\right) + 2 \]

   3. metadata-eval 99.8%

      \[\leadsto \frac{1}{y \cdot \color{blue}{0.25}} \cdot \left(x - z\right) + 2 \]

   4. associate-*l/ 100.0%

      \[\leadsto \color{blue}{\frac{1 \cdot \left(x - z\right)}{y \cdot 0.25}} + 2 \]

   5. *-un-lft-identity 100.0%

      \[\leadsto \frac{\color{blue}{x - z}}{y \cdot 0.25} + 2 \]

6. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\frac{x - z}{y \cdot 0.25}} + 2 \]
7. Add Preprocessing

Alternative 2: 80.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+172} \lor \neg \left(z \leq -4.2 \cdot 10^{+103}\right) \land \left(z \leq -1.1 \cdot 10^{+68} \lor \neg \left(z \leq 8 \cdot 10^{+106}\right)\right):\\ \;\;\;\;1 + -4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.9e+172)
         (and (not (<= z -4.2e+103)) (or (<= z -1.1e+68) (not (<= z 8e+106)))))
   (+ 1.0 (* -4.0 (/ z y)))
   (+ 2.0 (* 4.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.9e+172) || (!(z <= -4.2e+103) && ((z <= -1.1e+68) || !(z <= 8e+106)))) {
		tmp = 1.0 + (-4.0 * (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 <= (-1.9d+172)) .or. (.not. (z <= (-4.2d+103))) .and. (z <= (-1.1d+68)) .or. (.not. (z <= 8d+106))) then
        tmp = 1.0d0 + ((-4.0d0) * (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 <= -1.9e+172) || (!(z <= -4.2e+103) && ((z <= -1.1e+68) || !(z <= 8e+106)))) {
		tmp = 1.0 + (-4.0 * (z / y));
	} else {
		tmp = 2.0 + (4.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.9e+172) or (not (z <= -4.2e+103) and ((z <= -1.1e+68) or not (z <= 8e+106))):
		tmp = 1.0 + (-4.0 * (z / y))
	else:
		tmp = 2.0 + (4.0 * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.9e+172) || (!(z <= -4.2e+103) && ((z <= -1.1e+68) || !(z <= 8e+106))))
		tmp = Float64(1.0 + Float64(-4.0 * Float64(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 <= -1.9e+172) || (~((z <= -4.2e+103)) && ((z <= -1.1e+68) || ~((z <= 8e+106)))))
		tmp = 1.0 + (-4.0 * (z / y));
	else
		tmp = 2.0 + (4.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.9e+172], And[N[Not[LessEqual[z, -4.2e+103]], $MachinePrecision], Or[LessEqual[z, -1.1e+68], N[Not[LessEqual[z, 8e+106]], $MachinePrecision]]]], N[(1.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.89999999999999985e172 or -4.2000000000000003e103 < z < -1.09999999999999994e68 or 8.00000000000000073e106 < z

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 80.4%

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

      1. *-commutative 80.4%

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

   5. Simplified 80.4%

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

   if -1.89999999999999985e172 < z < -4.2000000000000003e103 or -1.09999999999999994e68 < z < 8.00000000000000073e106

   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. +-commutative 100.0%

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

      2. associate-*l/ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. associate-+l+ 99.8%

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

      8. associate-*l/ 99.8%

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

      9. *-commutative 99.8%

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

      10. associate-*l* 99.8%

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

      11. metadata-eval 99.8%

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

      12. *-rgt-identity 99.8%

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

      13. *-inverses 99.8%

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

      14. metadata-eval 99.8%

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

   3. Simplified 99.8%

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

      1. clear-num 99.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{4}}} \cdot \left(x - z\right) + 2 \]

      2. div-inv 99.8%

         \[\leadsto \frac{1}{\color{blue}{y \cdot \frac{1}{4}}} \cdot \left(x - z\right) + 2 \]

      3. metadata-eval 99.8%

         \[\leadsto \frac{1}{y \cdot \color{blue}{0.25}} \cdot \left(x - z\right) + 2 \]

      4. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(x - z\right)}{y \cdot 0.25}} + 2 \]

      5. *-un-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{x - z}}{y \cdot 0.25} + 2 \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{x - z}{y \cdot 0.25}} + 2 \]

   7. Taylor expanded in x around inf 83.3%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+172} \lor \neg \left(z \leq -4.2 \cdot 10^{+103}\right) \land \left(z \leq -1.1 \cdot 10^{+68} \lor \neg \left(z \leq 8 \cdot 10^{+106}\right)\right):\\ \;\;\;\;1 + -4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 57.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \frac{x \cdot 4}{y}\\ t_1 := 1 + -4 \cdot \frac{z}{y}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-177}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-138}:\\ \;\;\;\;2\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (/ (* x 4.0) y))) (t_1 (+ 1.0 (* -4.0 (/ z y)))))
   (if (<= z -1.15e+68)
     t_1
     (if (<= z 1.02e-177)
       t_0
       (if (<= z 4.8e-138) 2.0 (if (<= z 2.6e-20) t_0 t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + ((x * 4.0) / y);
	double t_1 = 1.0 + (-4.0 * (z / y));
	double tmp;
	if (z <= -1.15e+68) {
		tmp = t_1;
	} else if (z <= 1.02e-177) {
		tmp = t_0;
	} else if (z <= 4.8e-138) {
		tmp = 2.0;
	} else if (z <= 2.6e-20) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + ((x * 4.0d0) / y)
    t_1 = 1.0d0 + ((-4.0d0) * (z / y))
    if (z <= (-1.15d+68)) then
        tmp = t_1
    else if (z <= 1.02d-177) then
        tmp = t_0
    else if (z <= 4.8d-138) then
        tmp = 2.0d0
    else if (z <= 2.6d-20) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + ((x * 4.0) / y);
	double t_1 = 1.0 + (-4.0 * (z / y));
	double tmp;
	if (z <= -1.15e+68) {
		tmp = t_1;
	} else if (z <= 1.02e-177) {
		tmp = t_0;
	} else if (z <= 4.8e-138) {
		tmp = 2.0;
	} else if (z <= 2.6e-20) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + ((x * 4.0) / y)
	t_1 = 1.0 + (-4.0 * (z / y))
	tmp = 0
	if z <= -1.15e+68:
		tmp = t_1
	elif z <= 1.02e-177:
		tmp = t_0
	elif z <= 4.8e-138:
		tmp = 2.0
	elif z <= 2.6e-20:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(Float64(x * 4.0) / y))
	t_1 = Float64(1.0 + Float64(-4.0 * Float64(z / y)))
	tmp = 0.0
	if (z <= -1.15e+68)
		tmp = t_1;
	elseif (z <= 1.02e-177)
		tmp = t_0;
	elseif (z <= 4.8e-138)
		tmp = 2.0;
	elseif (z <= 2.6e-20)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + ((x * 4.0) / y);
	t_1 = 1.0 + (-4.0 * (z / y));
	tmp = 0.0;
	if (z <= -1.15e+68)
		tmp = t_1;
	elseif (z <= 1.02e-177)
		tmp = t_0;
	elseif (z <= 4.8e-138)
		tmp = 2.0;
	elseif (z <= 2.6e-20)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(N[(x * 4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.15e+68], t$95$1, If[LessEqual[z, 1.02e-177], t$95$0, If[LessEqual[z, 4.8e-138], 2.0, If[LessEqual[z, 2.6e-20], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.15e68 or 2.59999999999999995e-20 < z

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 68.4%

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

      1. *-commutative 68.4%

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

   5. Simplified 68.4%

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

   if -1.15e68 < z < 1.01999999999999997e-177 or 4.7999999999999998e-138 < z < 2.59999999999999995e-20

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 57.0%

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

      1. *-commutative 57.0%

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

      2. associate-*l/ 57.0%

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

   5. Simplified 57.0%

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

   if 1.01999999999999997e-177 < z < 4.7999999999999998e-138

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 90.1%

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

4. Final simplification 63.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{+68}:\\ \;\;\;\;1 + -4 \cdot \frac{z}{y}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{-177}:\\ \;\;\;\;1 + \frac{x \cdot 4}{y}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{-138}:\\ \;\;\;\;2\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-20}:\\ \;\;\;\;1 + \frac{x \cdot 4}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot \frac{4}{y}\\ t_1 := 1 + -4 \cdot \frac{z}{y}\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-178}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-138}:\\ \;\;\;\;2\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x (/ 4.0 y)))) (t_1 (+ 1.0 (* -4.0 (/ z y)))))
   (if (<= z -9.5e+67)
     t_1
     (if (<= z 7.5e-178)
       t_0
       (if (<= z 1.95e-138) 2.0 (if (<= z 2.8e-20) t_0 t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (x * (4.0 / y));
	double t_1 = 1.0 + (-4.0 * (z / y));
	double tmp;
	if (z <= -9.5e+67) {
		tmp = t_1;
	} else if (z <= 7.5e-178) {
		tmp = t_0;
	} else if (z <= 1.95e-138) {
		tmp = 2.0;
	} else if (z <= 2.8e-20) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (x * (4.0d0 / y))
    t_1 = 1.0d0 + ((-4.0d0) * (z / y))
    if (z <= (-9.5d+67)) then
        tmp = t_1
    else if (z <= 7.5d-178) then
        tmp = t_0
    else if (z <= 1.95d-138) then
        tmp = 2.0d0
    else if (z <= 2.8d-20) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (x * (4.0 / y));
	double t_1 = 1.0 + (-4.0 * (z / y));
	double tmp;
	if (z <= -9.5e+67) {
		tmp = t_1;
	} else if (z <= 7.5e-178) {
		tmp = t_0;
	} else if (z <= 1.95e-138) {
		tmp = 2.0;
	} else if (z <= 2.8e-20) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (x * (4.0 / y))
	t_1 = 1.0 + (-4.0 * (z / y))
	tmp = 0
	if z <= -9.5e+67:
		tmp = t_1
	elif z <= 7.5e-178:
		tmp = t_0
	elif z <= 1.95e-138:
		tmp = 2.0
	elif z <= 2.8e-20:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(x * Float64(4.0 / y)))
	t_1 = Float64(1.0 + Float64(-4.0 * Float64(z / y)))
	tmp = 0.0
	if (z <= -9.5e+67)
		tmp = t_1;
	elseif (z <= 7.5e-178)
		tmp = t_0;
	elseif (z <= 1.95e-138)
		tmp = 2.0;
	elseif (z <= 2.8e-20)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (x * (4.0 / y));
	t_1 = 1.0 + (-4.0 * (z / y));
	tmp = 0.0;
	if (z <= -9.5e+67)
		tmp = t_1;
	elseif (z <= 7.5e-178)
		tmp = t_0;
	elseif (z <= 1.95e-138)
		tmp = 2.0;
	elseif (z <= 2.8e-20)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.5e+67], t$95$1, If[LessEqual[z, 7.5e-178], t$95$0, If[LessEqual[z, 1.95e-138], 2.0, If[LessEqual[z, 2.8e-20], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.5000000000000002e67 or 2.8000000000000003e-20 < z

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 68.4%

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

      1. *-commutative 68.4%

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

   5. Simplified 68.4%

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

   if -9.5000000000000002e67 < z < 7.50000000000000019e-178 or 1.95e-138 < z < 2.8000000000000003e-20

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 57.0%

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

      1. associate-*r/ 57.0%

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

      2. associate-*l/ 56.8%

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

      3. *-commutative 56.8%

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

   5. Simplified 56.8%

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

   if 7.50000000000000019e-178 < z < 1.95e-138

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 90.1%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+67}:\\ \;\;\;\;1 + -4 \cdot \frac{z}{y}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-178}:\\ \;\;\;\;1 + x \cdot \frac{4}{y}\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{-138}:\\ \;\;\;\;2\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{-20}:\\ \;\;\;\;1 + x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + -4 \cdot \frac{z}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot \frac{4}{y}\\ t_1 := 1 + z \cdot \frac{-4}{y}\\ \mathbf{if}\;z \leq -9.5 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{-177}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{-146}:\\ \;\;\;\;2\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x (/ 4.0 y)))) (t_1 (+ 1.0 (* z (/ -4.0 y)))))
   (if (<= z -9.5e+67)
     t_1
     (if (<= z 2.9e-177)
       t_0
       (if (<= z 9.5e-146) 2.0 (if (<= z 1.6e-20) t_0 t_1))))))

C

double code(double x, double y, double z) {
	double t_0 = 1.0 + (x * (4.0 / y));
	double t_1 = 1.0 + (z * (-4.0 / y));
	double tmp;
	if (z <= -9.5e+67) {
		tmp = t_1;
	} else if (z <= 2.9e-177) {
		tmp = t_0;
	} else if (z <= 9.5e-146) {
		tmp = 2.0;
	} else if (z <= 1.6e-20) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_0 = 1.0d0 + (x * (4.0d0 / y))
    t_1 = 1.0d0 + (z * ((-4.0d0) / y))
    if (z <= (-9.5d+67)) then
        tmp = t_1
    else if (z <= 2.9d-177) then
        tmp = t_0
    else if (z <= 9.5d-146) then
        tmp = 2.0d0
    else if (z <= 1.6d-20) then
        tmp = t_0
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 1.0 + (x * (4.0 / y));
	double t_1 = 1.0 + (z * (-4.0 / y));
	double tmp;
	if (z <= -9.5e+67) {
		tmp = t_1;
	} else if (z <= 2.9e-177) {
		tmp = t_0;
	} else if (z <= 9.5e-146) {
		tmp = 2.0;
	} else if (z <= 1.6e-20) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 1.0 + (x * (4.0 / y))
	t_1 = 1.0 + (z * (-4.0 / y))
	tmp = 0
	if z <= -9.5e+67:
		tmp = t_1
	elif z <= 2.9e-177:
		tmp = t_0
	elif z <= 9.5e-146:
		tmp = 2.0
	elif z <= 1.6e-20:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(1.0 + Float64(x * Float64(4.0 / y)))
	t_1 = Float64(1.0 + Float64(z * Float64(-4.0 / y)))
	tmp = 0.0
	if (z <= -9.5e+67)
		tmp = t_1;
	elseif (z <= 2.9e-177)
		tmp = t_0;
	elseif (z <= 9.5e-146)
		tmp = 2.0;
	elseif (z <= 1.6e-20)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 1.0 + (x * (4.0 / y));
	t_1 = 1.0 + (z * (-4.0 / y));
	tmp = 0.0;
	if (z <= -9.5e+67)
		tmp = t_1;
	elseif (z <= 2.9e-177)
		tmp = t_0;
	elseif (z <= 9.5e-146)
		tmp = 2.0;
	elseif (z <= 1.6e-20)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(1.0 + N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.5e+67], t$95$1, If[LessEqual[z, 2.9e-177], t$95$0, If[LessEqual[z, 9.5e-146], 2.0, If[LessEqual[z, 1.6e-20], t$95$0, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.5000000000000002e67 or 1.59999999999999985e-20 < z

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 68.4%

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

      1. associate-*r/ 68.4%

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

      2. metadata-eval 68.4%

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

      3. associate-*r* 68.4%

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

      4. neg-mul-1 68.4%

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

      5. *-commutative 68.4%

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

      6. associate-*r/ 68.1%

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

      7. distribute-lft-neg-out 68.1%

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

      8. distribute-rgt-neg-in 68.1%

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

      9. distribute-neg-frac 68.1%

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

      10. metadata-eval 68.1%

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

   5. Simplified 68.1%

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

   if -9.5000000000000002e67 < z < 2.89999999999999997e-177 or 9.5000000000000005e-146 < z < 1.59999999999999985e-20

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 57.0%

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

      1. associate-*r/ 57.0%

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

      2. associate-*l/ 56.8%

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

      3. *-commutative 56.8%

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

   5. Simplified 56.8%

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

   if 2.89999999999999997e-177 < z < 9.5000000000000005e-146

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 90.1%

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

Alternative 6: 86.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+68} \lor \neg \left(z \leq 5.1 \cdot 10^{-20}\right):\\ \;\;\;\;2 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.02e+68) (not (<= z 5.1e-20)))
   (+ 2.0 (/ (* z -4.0) y))
   (+ 2.0 (* 4.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.02e+68) || !(z <= 5.1e-20)) {
		tmp = 2.0 + ((z * -4.0) / 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 <= (-1.02d+68)) .or. (.not. (z <= 5.1d-20))) then
        tmp = 2.0d0 + ((z * (-4.0d0)) / 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 <= -1.02e+68) || !(z <= 5.1e-20)) {
		tmp = 2.0 + ((z * -4.0) / y);
	} else {
		tmp = 2.0 + (4.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.02e+68) or not (z <= 5.1e-20):
		tmp = 2.0 + ((z * -4.0) / y)
	else:
		tmp = 2.0 + (4.0 * (x / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.02e+68) || !(z <= 5.1e-20))
		tmp = Float64(2.0 + Float64(Float64(z * -4.0) / 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 <= -1.02e+68) || ~((z <= 5.1e-20)))
		tmp = 2.0 + ((z * -4.0) / y);
	else
		tmp = 2.0 + (4.0 * (x / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.02e+68], N[Not[LessEqual[z, 5.1e-20]], $MachinePrecision]], N[(2.0 + N[(N[(z * -4.0), $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 < -1.02e68 or 5.10000000000000019e-20 < 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. +-commutative 100.0%

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

      2. associate-*l/ 99.7%

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

      3. +-commutative 99.7%

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

      4. associate--l+ 99.7%

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

      5. +-commutative 99.7%

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

      6. distribute-lft-in 98.8%

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

      7. associate-+l+ 98.8%

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

      8. associate-*l/ 99.7%

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

      9. *-commutative 99.7%

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

      10. associate-*l* 99.7%

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

      11. metadata-eval 99.7%

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

      12. *-rgt-identity 99.7%

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

      13. *-inverses 99.7%

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

      14. metadata-eval 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{4}}} \cdot \left(x - z\right) + 2 \]

      2. div-inv 99.7%

         \[\leadsto \frac{1}{\color{blue}{y \cdot \frac{1}{4}}} \cdot \left(x - z\right) + 2 \]

      3. metadata-eval 99.7%

         \[\leadsto \frac{1}{y \cdot \color{blue}{0.25}} \cdot \left(x - z\right) + 2 \]

      4. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(x - z\right)}{y \cdot 0.25}} + 2 \]

      5. *-un-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{x - z}}{y \cdot 0.25} + 2 \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{x - z}{y \cdot 0.25}} + 2 \]

   7. Taylor expanded in x around 0 85.6%

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

      1. associate-*r/ 85.6%

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

   9. Simplified 85.6%

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

   if -1.02e68 < z < 5.10000000000000019e-20

   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. +-commutative 100.0%

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

      2. associate-*l/ 99.8%

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

      3. +-commutative 99.8%

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

      4. associate--l+ 99.8%

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

      5. +-commutative 99.8%

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

      6. distribute-lft-in 99.8%

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

      7. associate-+l+ 99.9%

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

      8. associate-*l/ 99.9%

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

      9. *-commutative 99.9%

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

      10. associate-*l* 99.9%

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

      11. metadata-eval 99.9%

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

      12. *-rgt-identity 99.9%

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

      13. *-inverses 99.9%

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

      14. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{4}}} \cdot \left(x - z\right) + 2 \]

      2. div-inv 99.9%

         \[\leadsto \frac{1}{\color{blue}{y \cdot \frac{1}{4}}} \cdot \left(x - z\right) + 2 \]

      3. metadata-eval 99.9%

         \[\leadsto \frac{1}{y \cdot \color{blue}{0.25}} \cdot \left(x - z\right) + 2 \]

      4. associate-*l/ 100.0%

         \[\leadsto \color{blue}{\frac{1 \cdot \left(x - z\right)}{y \cdot 0.25}} + 2 \]

      5. *-un-lft-identity 100.0%

         \[\leadsto \frac{\color{blue}{x - z}}{y \cdot 0.25} + 2 \]

   6. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{x - z}{y \cdot 0.25}} + 2 \]

   7. Taylor expanded in x around inf 89.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.02 \cdot 10^{+68} \lor \neg \left(z \leq 5.1 \cdot 10^{-20}\right):\\ \;\;\;\;2 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;2 + 4 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 54.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+141} \lor \neg \left(x \leq 4.5 \cdot 10^{-9}\right):\\ \;\;\;\;1 + x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -9.5e+141) (not (<= x 4.5e-9))) (+ 1.0 (* x (/ 4.0 y))) 2.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -9.5e+141) || !(x <= 4.5e-9)) {
		tmp = 1.0 + (x * (4.0 / 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 ((x <= (-9.5d+141)) .or. (.not. (x <= 4.5d-9))) then
        tmp = 1.0d0 + (x * (4.0d0 / 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 ((x <= -9.5e+141) || !(x <= 4.5e-9)) {
		tmp = 1.0 + (x * (4.0 / y));
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -9.5e+141) or not (x <= 4.5e-9):
		tmp = 1.0 + (x * (4.0 / y))
	else:
		tmp = 2.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -9.5e+141) || !(x <= 4.5e-9))
		tmp = Float64(1.0 + Float64(x * Float64(4.0 / y)));
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -9.5e+141) || ~((x <= 4.5e-9)))
		tmp = 1.0 + (x * (4.0 / y));
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -9.5e+141], N[Not[LessEqual[x, 4.5e-9]], $MachinePrecision]], N[(1.0 + N[(x * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0]

Derivation

1. Split input into 2 regimes

2. if x < -9.49999999999999974e141 or 4.49999999999999976e-9 < x

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 71.1%

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

      1. associate-*r/ 71.1%

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

      2. associate-*l/ 70.9%

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

      3. *-commutative 70.9%

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

   5. Simplified 70.9%

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

   if -9.49999999999999974e141 < x < 4.49999999999999976e-9

   1. Initial program 100.0%

      \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 45.7%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{+141} \lor \neg \left(x \leq 4.5 \cdot 10^{-9}\right):\\ \;\;\;\;1 + x \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(2.0 + N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $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. +-commutative 100.0%

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

   2. associate-*l/ 99.8%

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

   3. +-commutative 99.8%

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

   4. associate--l+ 99.8%

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

   5. +-commutative 99.8%

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

   6. distribute-lft-in 99.4%

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

   7. associate-+l+ 99.4%

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

   8. associate-*l/ 99.8%

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

   9. *-commutative 99.8%

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

   10. associate-*l* 99.8%

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

   11. metadata-eval 99.8%

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

   12. *-rgt-identity 99.8%

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

   13. *-inverses 99.8%

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

   14. metadata-eval 99.8%

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

3. Simplified 99.8%

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

5. Final simplification 99.8%

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

Alternative 9: 34.8% 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. Add Preprocessing

3. Taylor expanded in y around inf 33.9%

   \[\leadsto \color{blue}{2} \]
4. Add Preprocessing

Alternative 10: 8.2% accurate, 13.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := 1.0

Derivation

1. Initial program 100.0%

   \[1 + \frac{4 \cdot \left(\left(x + y \cdot 0.25\right) - z\right)}{y} \]
2. Add Preprocessing

3. Taylor expanded in x around inf 38.8%

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

   1. associate-*r/ 38.8%

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

   2. associate-*l/ 38.7%

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

   3. *-commutative 38.7%

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

5. Simplified 38.7%

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

6. Taylor expanded in x around 0 8.2%

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

Reproduce

herbie shell --seed 2024085 
(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)))