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

Percentage Accurate: 99.7% → 100.0%

Time: 6.9s

Alternatives: 9

Speedup: 1.4×

• 20.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 1.0d0 + ((4.0d0 * ((x + (y * 0.75d0)) - z)) / y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. +-commutative 99.9%

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

   2. associate-/l* 99.9%

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

   3. fma-define 99.9%

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

   4. associate--l+ 99.9%

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

   5. +-commutative 99.9%

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

   6. remove-double-neg 99.9%

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

   7. sub-neg 99.9%

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

   8. associate--r+ 99.9%

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

   9. div-sub 99.9%

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

   10. sub-neg 99.9%

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

   11. associate-*l/ 100.0%

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

   12. *-inverses 100.0%

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

   13. metadata-eval 100.0%

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

   14. distribute-frac-neg2 100.0%

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

   15. remove-double-neg 100.0%

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

   16. distribute-neg-out 100.0%

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

   17. +-commutative 100.0%

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

   18. sub-neg 100.0%

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

   19. distribute-frac-neg 100.0%

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

   20. distribute-frac-neg2 100.0%

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

   21. remove-double-neg 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 53.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y}\\ t_1 := -4 \cdot \frac{z}{y}\\ \mathbf{if}\;x \leq -4.5 \cdot 10^{-5}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-160}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-199}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-242}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-174}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* 4.0 (/ x y))) (t_1 (* -4.0 (/ z y))))
   (if (<= x -4.5e-5)
     t_0
     (if (<= x -6.2e-160)
       t_1
       (if (<= x -1.05e-199)
         4.0
         (if (<= x 6e-242)
           t_1
           (if (<= x 7.8e-174) 4.0 (if (<= x 8.5e-17) t_1 t_0))))))))

C

double code(double x, double y, double z) {
	double t_0 = 4.0 * (x / y);
	double t_1 = -4.0 * (z / y);
	double tmp;
	if (x <= -4.5e-5) {
		tmp = t_0;
	} else if (x <= -6.2e-160) {
		tmp = t_1;
	} else if (x <= -1.05e-199) {
		tmp = 4.0;
	} else if (x <= 6e-242) {
		tmp = t_1;
	} else if (x <= 7.8e-174) {
		tmp = 4.0;
	} else if (x <= 8.5e-17) {
		tmp = t_1;
	} 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) :: t_1
    real(8) :: tmp
    t_0 = 4.0d0 * (x / y)
    t_1 = (-4.0d0) * (z / y)
    if (x <= (-4.5d-5)) then
        tmp = t_0
    else if (x <= (-6.2d-160)) then
        tmp = t_1
    else if (x <= (-1.05d-199)) then
        tmp = 4.0d0
    else if (x <= 6d-242) then
        tmp = t_1
    else if (x <= 7.8d-174) then
        tmp = 4.0d0
    else if (x <= 8.5d-17) then
        tmp = t_1
    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 t_1 = -4.0 * (z / y);
	double tmp;
	if (x <= -4.5e-5) {
		tmp = t_0;
	} else if (x <= -6.2e-160) {
		tmp = t_1;
	} else if (x <= -1.05e-199) {
		tmp = 4.0;
	} else if (x <= 6e-242) {
		tmp = t_1;
	} else if (x <= 7.8e-174) {
		tmp = 4.0;
	} else if (x <= 8.5e-17) {
		tmp = t_1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 4.0 * (x / y)
	t_1 = -4.0 * (z / y)
	tmp = 0
	if x <= -4.5e-5:
		tmp = t_0
	elif x <= -6.2e-160:
		tmp = t_1
	elif x <= -1.05e-199:
		tmp = 4.0
	elif x <= 6e-242:
		tmp = t_1
	elif x <= 7.8e-174:
		tmp = 4.0
	elif x <= 8.5e-17:
		tmp = t_1
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(4.0 * Float64(x / y))
	t_1 = Float64(-4.0 * Float64(z / y))
	tmp = 0.0
	if (x <= -4.5e-5)
		tmp = t_0;
	elseif (x <= -6.2e-160)
		tmp = t_1;
	elseif (x <= -1.05e-199)
		tmp = 4.0;
	elseif (x <= 6e-242)
		tmp = t_1;
	elseif (x <= 7.8e-174)
		tmp = 4.0;
	elseif (x <= 8.5e-17)
		tmp = t_1;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 4.0 * (x / y);
	t_1 = -4.0 * (z / y);
	tmp = 0.0;
	if (x <= -4.5e-5)
		tmp = t_0;
	elseif (x <= -6.2e-160)
		tmp = t_1;
	elseif (x <= -1.05e-199)
		tmp = 4.0;
	elseif (x <= 6e-242)
		tmp = t_1;
	elseif (x <= 7.8e-174)
		tmp = 4.0;
	elseif (x <= 8.5e-17)
		tmp = t_1;
	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]}, Block[{t$95$1 = N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -4.5e-5], t$95$0, If[LessEqual[x, -6.2e-160], t$95$1, If[LessEqual[x, -1.05e-199], 4.0, If[LessEqual[x, 6e-242], t$95$1, If[LessEqual[x, 7.8e-174], 4.0, If[LessEqual[x, 8.5e-17], t$95$1, t$95$0]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.50000000000000028e-5 or 8.5e-17 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 57.5%

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

   if -4.50000000000000028e-5 < x < -6.2000000000000001e-160 or -1.05000000000000001e-199 < x < 6e-242 or 7.7999999999999997e-174 < x < 8.5e-17

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in z around inf 62.7%

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

   if -6.2000000000000001e-160 < x < -1.05000000000000001e-199 or 6e-242 < x < 7.7999999999999997e-174

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 82.4%

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

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4.5 \cdot 10^{-5}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \mathbf{elif}\;x \leq -6.2 \cdot 10^{-160}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-199}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 6 \cdot 10^{-242}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-174}:\\ \;\;\;\;4\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{-17}:\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;4 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-32} \lor \neg \left(y \leq 6.5 \cdot 10^{+18}\right):\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x - z\right) \cdot \frac{4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.1e-32) (not (<= y 6.5e+18)))
   (+ 4.0 (/ (* z -4.0) y))
   (+ 1.0 (* (- x z) (/ 4.0 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.1e-32) || !(y <= 6.5e+18)) {
		tmp = 4.0 + ((z * -4.0) / y);
	} else {
		tmp = 1.0 + ((x - z) * (4.0 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-3.1d-32)) .or. (.not. (y <= 6.5d+18))) then
        tmp = 4.0d0 + ((z * (-4.0d0)) / y)
    else
        tmp = 1.0d0 + ((x - z) * (4.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.1e-32) || !(y <= 6.5e+18)) {
		tmp = 4.0 + ((z * -4.0) / y);
	} else {
		tmp = 1.0 + ((x - z) * (4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.1e-32) or not (y <= 6.5e+18):
		tmp = 4.0 + ((z * -4.0) / y)
	else:
		tmp = 1.0 + ((x - z) * (4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.1e-32) || !(y <= 6.5e+18))
		tmp = Float64(4.0 + Float64(Float64(z * -4.0) / y));
	else
		tmp = Float64(1.0 + Float64(Float64(x - z) * Float64(4.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.1e-32) || ~((y <= 6.5e+18)))
		tmp = 4.0 + ((z * -4.0) / y);
	else
		tmp = 1.0 + ((x - z) * (4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.1e-32], N[Not[LessEqual[y, 6.5e+18]], $MachinePrecision]], N[(4.0 + N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.10000000000000011e-32 or 6.5e18 < y

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. associate--r+ 99.9%

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

      9. div-sub 99.9%

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

      10. sub-neg 99.9%

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

      11. associate-*l/ 100.0%

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

      12. *-inverses 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-frac-neg2 100.0%

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

      15. remove-double-neg 100.0%

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

      16. distribute-neg-out 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. distribute-frac-neg2 100.0%

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

      21. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 84.4%

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

      1. sub-neg 84.4%

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

      2. distribute-lft-in 84.4%

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

      3. metadata-eval 84.4%

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

      4. associate-+r+ 84.4%

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

      5. metadata-eval 84.4%

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

      6. neg-mul-1 84.4%

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

      7. associate-*r* 84.4%

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

      8. metadata-eval 84.4%

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

      9. *-commutative 84.4%

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

      10. associate-*l/ 84.4%

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

   7. Simplified 84.4%

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

   if -3.10000000000000011e-32 < y < 6.5e18

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 94.1%

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

      1. *-lft-identity 94.1%

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

      2. associate-*l/ 93.8%

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

      3. associate-*r* 93.8%

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

      4. associate-*r/ 93.8%

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

      5. metadata-eval 93.8%

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

   5. Simplified 93.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-32} \lor \neg \left(y \leq 6.5 \cdot 10^{+18}\right):\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(x - z\right) \cdot \frac{4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+79} \lor \neg \left(z \leq 2.9 \cdot 10^{+47}\right):\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -8.2e+79) (not (<= z 2.9e+47)))
   (* (- x z) (/ 4.0 y))
   (+ 4.0 (/ (* 4.0 x) y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -8.2e+79) || !(z <= 2.9e+47)) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = 4.0 + ((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 <= (-8.2d+79)) .or. (.not. (z <= 2.9d+47))) then
        tmp = (x - z) * (4.0d0 / y)
    else
        tmp = 4.0d0 + ((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 <= -8.2e+79) || !(z <= 2.9e+47)) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = 4.0 + ((4.0 * x) / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -8.2e+79) or not (z <= 2.9e+47):
		tmp = (x - z) * (4.0 / y)
	else:
		tmp = 4.0 + ((4.0 * x) / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -8.2e+79) || !(z <= 2.9e+47))
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	else
		tmp = Float64(4.0 + Float64(Float64(4.0 * x) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -8.2e+79) || ~((z <= 2.9e+47)))
		tmp = (x - z) * (4.0 / y);
	else
		tmp = 4.0 + ((4.0 * x) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -8.2e+79], N[Not[LessEqual[z, 2.9e+47]], $MachinePrecision]], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(N[(4.0 * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.2e79 or 2.8999999999999998e47 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. distribute-lft-out 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in y around 0 86.5%

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

      1. *-commutative 86.5%

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

      2. associate-*l/ 86.5%

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

      3. associate-/l* 86.2%

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

   8. Simplified 86.2%

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

   if -8.2e79 < z < 2.8999999999999998e47

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. associate--r+ 99.9%

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

      9. div-sub 99.9%

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

      10. sub-neg 99.9%

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

      11. associate-*l/ 100.0%

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

      12. *-inverses 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-frac-neg2 100.0%

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

      15. remove-double-neg 100.0%

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

      16. distribute-neg-out 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. distribute-frac-neg2 100.0%

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

      21. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in z around 0 88.3%

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

      1. distribute-lft-in 88.3%

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

      2. metadata-eval 88.3%

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

      3. associate-+r+ 88.3%

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

      4. metadata-eval 88.3%

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

      5. *-commutative 88.3%

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

      6. associate-*l/ 88.3%

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

   7. Simplified 88.3%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+79} \lor \neg \left(z \leq 2.9 \cdot 10^{+47}\right):\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + \frac{4 \cdot x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-32} \lor \neg \left(y \leq 37000\right):\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -3.1e-32) (not (<= y 37000.0)))
   (+ 4.0 (/ (* z -4.0) y))
   (* (- x z) (/ 4.0 y))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.1e-32) || !(y <= 37000.0)) {
		tmp = 4.0 + ((z * -4.0) / y);
	} else {
		tmp = (x - z) * (4.0 / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= (-3.1d-32)) .or. (.not. (y <= 37000.0d0))) then
        tmp = 4.0d0 + ((z * (-4.0d0)) / y)
    else
        tmp = (x - z) * (4.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((y <= -3.1e-32) || !(y <= 37000.0)) {
		tmp = 4.0 + ((z * -4.0) / y);
	} else {
		tmp = (x - z) * (4.0 / y);
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (y <= -3.1e-32) or not (y <= 37000.0):
		tmp = 4.0 + ((z * -4.0) / y)
	else:
		tmp = (x - z) * (4.0 / y)
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -3.1e-32) || !(y <= 37000.0))
		tmp = Float64(4.0 + Float64(Float64(z * -4.0) / y));
	else
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((y <= -3.1e-32) || ~((y <= 37000.0)))
		tmp = 4.0 + ((z * -4.0) / y);
	else
		tmp = (x - z) * (4.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -3.1e-32], N[Not[LessEqual[y, 37000.0]], $MachinePrecision]], N[(4.0 + N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.10000000000000011e-32 or 37000 < y

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. fma-define 99.9%

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

      4. associate--l+ 99.9%

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

      5. +-commutative 99.9%

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

      6. remove-double-neg 99.9%

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

      7. sub-neg 99.9%

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

      8. associate--r+ 99.9%

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

      9. div-sub 99.9%

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

      10. sub-neg 99.9%

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

      11. associate-*l/ 100.0%

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

      12. *-inverses 100.0%

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

      13. metadata-eval 100.0%

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

      14. distribute-frac-neg2 100.0%

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

      15. remove-double-neg 100.0%

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

      16. distribute-neg-out 100.0%

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

      17. +-commutative 100.0%

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

      18. sub-neg 100.0%

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

      19. distribute-frac-neg 100.0%

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

      20. distribute-frac-neg2 100.0%

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

      21. remove-double-neg 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 83.6%

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

      1. sub-neg 83.6%

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

      2. distribute-lft-in 83.6%

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

      3. metadata-eval 83.6%

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

      4. associate-+r+ 83.6%

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

      5. metadata-eval 83.6%

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

      6. neg-mul-1 83.6%

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

      7. associate-*r* 83.6%

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

      8. metadata-eval 83.6%

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

      9. *-commutative 83.6%

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

      10. associate-*l/ 83.6%

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

   7. Simplified 83.6%

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

   if -3.10000000000000011e-32 < y < 37000

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 94.4%

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

      1. *-commutative 94.4%

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

      2. associate-*l/ 94.4%

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

      3. associate-/l* 94.1%

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

   8. Simplified 94.1%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.1 \cdot 10^{-32} \lor \neg \left(y \leq 37000\right):\\ \;\;\;\;4 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+91}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+179}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.2e+91) 4.0 (if (<= y 4.9e+179) (* (- x z) (/ 4.0 y)) 4.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.2e+91) {
		tmp = 4.0;
	} else if (y <= 4.9e+179) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-8.2d+91)) then
        tmp = 4.0d0
    else if (y <= 4.9d+179) then
        tmp = (x - z) * (4.0d0 / y)
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.2e+91) {
		tmp = 4.0;
	} else if (y <= 4.9e+179) {
		tmp = (x - z) * (4.0 / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.2e+91:
		tmp = 4.0
	elif y <= 4.9e+179:
		tmp = (x - z) * (4.0 / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.2e+91)
		tmp = 4.0;
	elseif (y <= 4.9e+179)
		tmp = Float64(Float64(x - z) * Float64(4.0 / y));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.2e+91)
		tmp = 4.0;
	elseif (y <= 4.9e+179)
		tmp = (x - z) * (4.0 / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.2e+91], 4.0, If[LessEqual[y, 4.9e+179], N[(N[(x - z), $MachinePrecision] * N[(4.0 / y), $MachinePrecision]), $MachinePrecision], 4.0]]

Derivation

1. Split input into 2 regimes

2. if y < -8.2000000000000005e91 or 4.8999999999999999e179 < y

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 73.4%

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

   if -8.2000000000000005e91 < y < 4.8999999999999999e179

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

      1. distribute-lft-out 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in y around 0 83.9%

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

      1. *-commutative 83.9%

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

      2. associate-*l/ 83.9%

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

      3. associate-/l* 83.7%

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

   8. Simplified 83.7%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.2 \cdot 10^{+91}:\\ \;\;\;\;4\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+179}:\\ \;\;\;\;\left(x - z\right) \cdot \frac{4}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 53.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+54} \lor \neg \left(z \leq 2.6 \cdot 10^{+48}\right):\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -1.25e+54) (not (<= z 2.6e+48))) (* -4.0 (/ z y)) 4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e+54) || !(z <= 2.6e+48)) {
		tmp = -4.0 * (z / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((z <= (-1.25d+54)) .or. (.not. (z <= 2.6d+48))) then
        tmp = (-4.0d0) * (z / y)
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((z <= -1.25e+54) || !(z <= 2.6e+48)) {
		tmp = -4.0 * (z / y);
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -1.25e+54) or not (z <= 2.6e+48):
		tmp = -4.0 * (z / y)
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((z <= -1.25e+54) || !(z <= 2.6e+48))
		tmp = Float64(-4.0 * Float64(z / y));
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -1.25e+54) || ~((z <= 2.6e+48)))
		tmp = -4.0 * (z / y);
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[z, -1.25e+54], N[Not[LessEqual[z, 2.6e+48]], $MachinePrecision]], N[(-4.0 * N[(z / y), $MachinePrecision]), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if z < -1.25000000000000001e54 or 2.59999999999999995e48 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 99.9%

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

      1. distribute-lft-out 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in z around inf 71.1%

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

   if -1.25000000000000001e54 < z < 2.59999999999999995e48

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 43.7%

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

4. Final simplification 55.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+54} \lor \neg \left(z \leq 2.6 \cdot 10^{+48}\right):\\ \;\;\;\;-4 \cdot \frac{z}{y}\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.8% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around 0 100.0%

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

   1. distribute-lft-out 100.0%

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

5. Simplified 100.0%

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

6. Final simplification 100.0%

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

Alternative 9: 34.8% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ 4 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 4.0)

C

double code(double x, double y, double z) {
	return 4.0;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 4.0d0
end function

Java

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

Python

def code(x, y, z):
	return 4.0

Julia

function code(x, y, z)
	return 4.0
end

MATLAB

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

Wolfram

code[x_, y_, z_] := 4.0

Derivation

1. Initial program 99.9%

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

3. Taylor expanded in y around inf 31.4%

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

4. Final simplification 31.4%

   \[\leadsto 4 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, A"
  :precision binary64
  (+ 1.0 (/ (* 4.0 (- (+ x (* y 0.75)) z)) y)))