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

Percentage Accurate: 99.8% → 99.9%

Time: 5.9s

Alternatives: 6

Speedup: 1.4×

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.2%

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

   1. +-commutative 99.2%

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

   2. associate-*l/ 99.7%

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

   3. *-commutative 99.7%

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

   4. fma-def 99.7%

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

   5. +-commutative 99.7%

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

   6. associate--l+ 99.7%

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

   7. +-commutative 99.7%

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

   8. associate-+l- 99.7%

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

   9. sub-neg 99.7%

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

   10. remove-double-neg 99.7%

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

   11. distribute-neg-in 99.7%

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

   12. +-commutative 99.7%

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

   13. distribute-neg-in 99.7%

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

   14. distribute-lft-neg-out 99.7%

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

   15. sub-neg 99.7%

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

   16. distribute-lft-neg-out 99.7%

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

   17. distribute-rgt-neg-in 99.7%

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

   18. fma-neg 99.7%

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

   19. metadata-eval 99.7%

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

   20. remove-double-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in y around 0 100.0%

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

   1. associate-*r/ 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 56.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 4 \cdot \frac{x}{y} + 1\\ t_1 := 1 + \frac{z \cdot -4}{y}\\ \mathbf{if}\;z \leq -17500000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-41}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-145}:\\ \;\;\;\;4\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{-301}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-152}:\\ \;\;\;\;4\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+84}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (+ (* 4.0 (/ x y)) 1.0)) (t_1 (+ 1.0 (/ (* z -4.0) y))))
   (if (<= z -17500000000000.0)
     t_1
     (if (<= z -1.4e-41)
       t_0
       (if (<= z -7e-145)
         4.0
         (if (<= z 1.06e-301)
           t_0
           (if (<= z 4e-152) 4.0 (if (<= z 1.45e+84) t_0 t_1))))))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * (x / y)) + 1.0;
	double t_1 = 1.0 + ((z * -4.0) / y);
	double tmp;
	if (z <= -17500000000000.0) {
		tmp = t_1;
	} else if (z <= -1.4e-41) {
		tmp = t_0;
	} else if (z <= -7e-145) {
		tmp = 4.0;
	} else if (z <= 1.06e-301) {
		tmp = t_0;
	} else if (z <= 4e-152) {
		tmp = 4.0;
	} else if (z <= 1.45e+84) {
		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 = (4.0d0 * (x / y)) + 1.0d0
    t_1 = 1.0d0 + ((z * (-4.0d0)) / y)
    if (z <= (-17500000000000.0d0)) then
        tmp = t_1
    else if (z <= (-1.4d-41)) then
        tmp = t_0
    else if (z <= (-7d-145)) then
        tmp = 4.0d0
    else if (z <= 1.06d-301) then
        tmp = t_0
    else if (z <= 4d-152) then
        tmp = 4.0d0
    else if (z <= 1.45d+84) 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 = (4.0 * (x / y)) + 1.0;
	double t_1 = 1.0 + ((z * -4.0) / y);
	double tmp;
	if (z <= -17500000000000.0) {
		tmp = t_1;
	} else if (z <= -1.4e-41) {
		tmp = t_0;
	} else if (z <= -7e-145) {
		tmp = 4.0;
	} else if (z <= 1.06e-301) {
		tmp = t_0;
	} else if (z <= 4e-152) {
		tmp = 4.0;
	} else if (z <= 1.45e+84) {
		tmp = t_0;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * (x / y)) + 1.0
	t_1 = 1.0 + ((z * -4.0) / y)
	tmp = 0
	if z <= -17500000000000.0:
		tmp = t_1
	elif z <= -1.4e-41:
		tmp = t_0
	elif z <= -7e-145:
		tmp = 4.0
	elif z <= 1.06e-301:
		tmp = t_0
	elif z <= 4e-152:
		tmp = 4.0
	elif z <= 1.45e+84:
		tmp = t_0
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(x / y)) + 1.0)
	t_1 = Float64(1.0 + Float64(Float64(z * -4.0) / y))
	tmp = 0.0
	if (z <= -17500000000000.0)
		tmp = t_1;
	elseif (z <= -1.4e-41)
		tmp = t_0;
	elseif (z <= -7e-145)
		tmp = 4.0;
	elseif (z <= 1.06e-301)
		tmp = t_0;
	elseif (z <= 4e-152)
		tmp = 4.0;
	elseif (z <= 1.45e+84)
		tmp = t_0;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * (x / y)) + 1.0;
	t_1 = 1.0 + ((z * -4.0) / y);
	tmp = 0.0;
	if (z <= -17500000000000.0)
		tmp = t_1;
	elseif (z <= -1.4e-41)
		tmp = t_0;
	elseif (z <= -7e-145)
		tmp = 4.0;
	elseif (z <= 1.06e-301)
		tmp = t_0;
	elseif (z <= 4e-152)
		tmp = 4.0;
	elseif (z <= 1.45e+84)
		tmp = t_0;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -17500000000000.0], t$95$1, If[LessEqual[z, -1.4e-41], t$95$0, If[LessEqual[z, -7e-145], 4.0, If[LessEqual[z, 1.06e-301], t$95$0, If[LessEqual[z, 4e-152], 4.0, If[LessEqual[z, 1.45e+84], t$95$0, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.75e13 or 1.44999999999999994e84 < z

   1. Initial program 99.1%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. metadata-eval 67.1%

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

      3. associate-*r* 67.1%

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

      4. neg-mul-1 67.1%

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

      5. associate-*l/ 67.0%

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

      6. *-commutative 67.0%

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

      7. associate-*r/ 67.1%

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

      8. neg-mul-1 67.1%

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

      9. *-commutative 67.1%

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

      10. associate-*l* 67.1%

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

      11. metadata-eval 67.1%

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

   7. Simplified 67.1%

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

   if -1.75e13 < z < -1.4000000000000001e-41 or -6.99999999999999994e-145 < z < 1.06e-301 or 4.00000000000000026e-152 < z < 1.44999999999999994e84

   1. Initial program 98.7%

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

      1. associate-*l/ 99.6%

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

      2. +-commutative 99.6%

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

      3. fma-def 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 65.2%

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

   if -1.4000000000000001e-41 < z < -6.99999999999999994e-145 or 1.06e-301 < z < 4.00000000000000026e-152

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*l/ 99.7%

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

      3. *-commutative 99.7%

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

      4. fma-def 99.6%

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

      5. +-commutative 99.6%

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

      6. associate--l+ 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+l- 99.6%

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

      9. sub-neg 99.6%

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

      10. remove-double-neg 99.6%

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

      11. distribute-neg-in 99.6%

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

      12. +-commutative 99.6%

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

      13. distribute-neg-in 99.6%

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

      14. distribute-lft-neg-out 99.6%

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

      15. sub-neg 99.6%

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

      16. distribute-lft-neg-out 99.6%

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

      17. distribute-rgt-neg-in 99.6%

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

      18. fma-neg 99.6%

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

      19. metadata-eval 99.6%

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

      20. remove-double-neg 99.6%

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

   3. Simplified 99.6%

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

   5. Taylor expanded in y around inf 56.8%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -17500000000000:\\ \;\;\;\;1 + \frac{z \cdot -4}{y}\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-41}:\\ \;\;\;\;4 \cdot \frac{x}{y} + 1\\ \mathbf{elif}\;z \leq -7 \cdot 10^{-145}:\\ \;\;\;\;4\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{-301}:\\ \;\;\;\;4 \cdot \frac{x}{y} + 1\\ \mathbf{elif}\;z \leq 4 \cdot 10^{-152}:\\ \;\;\;\;4\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{+84}:\\ \;\;\;\;4 \cdot \frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{z \cdot -4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -17500000000000 \lor \neg \left(z \leq 3 \cdot 10^{+84} \lor \neg \left(z \leq 1.52 \cdot 10^{+147}\right) \land z \leq 1.18 \cdot 10^{+210}\right):\\ \;\;\;\;1 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= z -17500000000000.0)
         (not (or (<= z 3e+84) (and (not (<= z 1.52e+147)) (<= z 1.18e+210)))))
   (+ 1.0 (/ (* z -4.0) y))
   (+ 4.0 (* 4.0 (/ x y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((z <= -17500000000000.0) || !((z <= 3e+84) || (!(z <= 1.52e+147) && (z <= 1.18e+210)))) {
		tmp = 1.0 + ((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 <= (-17500000000000.0d0)) .or. (.not. (z <= 3d+84) .or. (.not. (z <= 1.52d+147)) .and. (z <= 1.18d+210))) then
        tmp = 1.0d0 + ((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 <= -17500000000000.0) || !((z <= 3e+84) || (!(z <= 1.52e+147) && (z <= 1.18e+210)))) {
		tmp = 1.0 + ((z * -4.0) / y);
	} else {
		tmp = 4.0 + (4.0 * (x / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (z <= -17500000000000.0) or not ((z <= 3e+84) or (not (z <= 1.52e+147) and (z <= 1.18e+210))):
		tmp = 1.0 + ((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 <= -17500000000000.0) || !((z <= 3e+84) || (!(z <= 1.52e+147) && (z <= 1.18e+210))))
		tmp = Float64(1.0 + Float64(Float64(z * -4.0) / y));
	else
		tmp = Float64(4.0 + Float64(4.0 * Float64(x / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((z <= -17500000000000.0) || ~(((z <= 3e+84) || (~((z <= 1.52e+147)) && (z <= 1.18e+210)))))
		tmp = 1.0 + ((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, -17500000000000.0], N[Not[Or[LessEqual[z, 3e+84], And[N[Not[LessEqual[z, 1.52e+147]], $MachinePrecision], LessEqual[z, 1.18e+210]]]], $MachinePrecision]], N[(1.0 + N[(N[(z * -4.0), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.75e13 or 2.99999999999999996e84 < z < 1.51999999999999992e147 or 1.18e210 < z

   1. Initial program 99.0%

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

      1. associate-*l/ 99.8%

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

      2. +-commutative 99.8%

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

      3. fma-def 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 71.6%

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

      1. associate-*r/ 71.6%

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

      2. metadata-eval 71.6%

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

      3. associate-*r* 71.6%

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

      4. neg-mul-1 71.6%

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

      5. associate-*l/ 71.6%

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

      6. *-commutative 71.6%

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

      7. associate-*r/ 71.6%

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

      8. neg-mul-1 71.6%

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

      9. *-commutative 71.6%

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

      10. associate-*l* 71.6%

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

      11. metadata-eval 71.6%

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

   7. Simplified 71.6%

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

   if -1.75e13 < z < 2.99999999999999996e84 or 1.51999999999999992e147 < z < 1.18e210

   1. Initial program 99.3%

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

      1. +-commutative 99.3%

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

      2. associate-*l/ 99.7%

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

      3. *-commutative 99.7%

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

      4. fma-def 99.7%

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

      5. +-commutative 99.7%

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

      6. associate--l+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l- 99.7%

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

      9. sub-neg 99.7%

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

      10. remove-double-neg 99.7%

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

      11. distribute-neg-in 99.7%

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

      12. +-commutative 99.7%

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

      13. distribute-neg-in 99.7%

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

      14. distribute-lft-neg-out 99.7%

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

      15. sub-neg 99.7%

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

      16. distribute-lft-neg-out 99.7%

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

      17. distribute-rgt-neg-in 99.7%

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

      18. fma-neg 99.7%

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

      19. metadata-eval 99.7%

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

      20. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 87.9%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -17500000000000 \lor \neg \left(z \leq 3 \cdot 10^{+84} \lor \neg \left(z \leq 1.52 \cdot 10^{+147}\right) \land z \leq 1.18 \cdot 10^{+210}\right):\\ \;\;\;\;1 + \frac{z \cdot -4}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+42} \lor \neg \left(x \leq 11800000\right):\\ \;\;\;\;4 \cdot \frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.25e+42) (not (<= x 11800000.0)))
   (+ (* 4.0 (/ x y)) 1.0)
   4.0))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.25e+42) || !(x <= 11800000.0)) {
		tmp = (4.0 * (x / y)) + 1.0;
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.25d+42)) .or. (.not. (x <= 11800000.0d0))) then
        tmp = (4.0d0 * (x / y)) + 1.0d0
    else
        tmp = 4.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.25e+42) || !(x <= 11800000.0)) {
		tmp = (4.0 * (x / y)) + 1.0;
	} else {
		tmp = 4.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.25e+42) or not (x <= 11800000.0):
		tmp = (4.0 * (x / y)) + 1.0
	else:
		tmp = 4.0
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.25e+42) || !(x <= 11800000.0))
		tmp = Float64(Float64(4.0 * Float64(x / y)) + 1.0);
	else
		tmp = 4.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.25e+42) || ~((x <= 11800000.0)))
		tmp = (4.0 * (x / y)) + 1.0;
	else
		tmp = 4.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.25e+42], N[Not[LessEqual[x, 11800000.0]], $MachinePrecision]], N[(N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision], 4.0]

Derivation

1. Split input into 2 regimes

2. if x < -1.25000000000000002e42 or 1.18e7 < x

   1. Initial program 99.2%

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

      1. associate-*l/ 99.7%

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

      2. +-commutative 99.7%

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

      3. fma-def 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in x around inf 71.3%

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

   if -1.25000000000000002e42 < x < 1.18e7

   1. Initial program 99.2%

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

      1. +-commutative 99.2%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

      4. fma-def 99.7%

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

      5. +-commutative 99.7%

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

      6. associate--l+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l- 99.7%

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

      9. sub-neg 99.7%

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

      10. remove-double-neg 99.7%

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

      11. distribute-neg-in 99.7%

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

      12. +-commutative 99.7%

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

      13. distribute-neg-in 99.7%

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

      14. distribute-lft-neg-out 99.7%

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

      15. sub-neg 99.7%

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

      16. distribute-lft-neg-out 99.7%

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

      17. distribute-rgt-neg-in 99.7%

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

      18. fma-neg 99.7%

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

      19. metadata-eval 99.7%

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

      20. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around inf 47.2%

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

4. Final simplification 58.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{+42} \lor \neg \left(x \leq 11800000\right):\\ \;\;\;\;4 \cdot \frac{x}{y} + 1\\ \mathbf{else}:\\ \;\;\;\;4\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+52} \lor \neg \left(x \leq 4.5 \cdot 10^{+155}\right):\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + z \cdot \frac{-4}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -1.12e+52) (not (<= x 4.5e+155)))
   (+ 4.0 (* 4.0 (/ x y)))
   (+ 4.0 (* z (/ -4.0 y)))))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.12e+52) || !(x <= 4.5e+155)) {
		tmp = 4.0 + (4.0 * (x / y));
	} else {
		tmp = 4.0 + (z * (-4.0 / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((x <= (-1.12d+52)) .or. (.not. (x <= 4.5d+155))) then
        tmp = 4.0d0 + (4.0d0 * (x / y))
    else
        tmp = 4.0d0 + (z * ((-4.0d0) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if ((x <= -1.12e+52) || !(x <= 4.5e+155)) {
		tmp = 4.0 + (4.0 * (x / y));
	} else {
		tmp = 4.0 + (z * (-4.0 / y));
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if (x <= -1.12e+52) or not (x <= 4.5e+155):
		tmp = 4.0 + (4.0 * (x / y))
	else:
		tmp = 4.0 + (z * (-4.0 / y))
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -1.12e+52) || !(x <= 4.5e+155))
		tmp = Float64(4.0 + Float64(4.0 * Float64(x / y)));
	else
		tmp = Float64(4.0 + Float64(z * Float64(-4.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if ((x <= -1.12e+52) || ~((x <= 4.5e+155)))
		tmp = 4.0 + (4.0 * (x / y));
	else
		tmp = 4.0 + (z * (-4.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -1.12e+52], N[Not[LessEqual[x, 4.5e+155]], $MachinePrecision]], N[(4.0 + N[(4.0 * N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(4.0 + N[(z * N[(-4.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.12000000000000002e52 or 4.49999999999999973e155 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-*l/ 99.6%

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

      3. *-commutative 99.6%

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

      4. fma-def 99.6%

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

      5. +-commutative 99.6%

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

      6. associate--l+ 99.6%

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

      7. +-commutative 99.6%

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

      8. associate-+l- 99.6%

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

      9. sub-neg 99.6%

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

      10. remove-double-neg 99.6%

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

      11. distribute-neg-in 99.6%

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

      12. +-commutative 99.6%

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

      13. distribute-neg-in 99.6%

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

      14. distribute-lft-neg-out 99.6%

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

      15. sub-neg 99.6%

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

      16. distribute-lft-neg-out 99.6%

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

      17. distribute-rgt-neg-in 99.6%

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

      18. fma-neg 99.7%

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

      19. metadata-eval 99.7%

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

      20. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 92.9%

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

   if -1.12000000000000002e52 < x < 4.49999999999999973e155

   1. Initial program 98.8%

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

      1. +-commutative 98.8%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

      4. fma-def 99.7%

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

      5. +-commutative 99.7%

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

      6. associate--l+ 99.7%

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

      7. +-commutative 99.7%

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

      8. associate-+l- 99.7%

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

      9. sub-neg 99.7%

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

      10. remove-double-neg 99.7%

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

      11. distribute-neg-in 99.7%

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

      12. +-commutative 99.7%

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

      13. distribute-neg-in 99.7%

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

      14. distribute-lft-neg-out 99.7%

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

      15. sub-neg 99.7%

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

      16. distribute-lft-neg-out 99.7%

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

      17. distribute-rgt-neg-in 99.7%

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

      18. fma-neg 99.7%

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

      19. metadata-eval 99.7%

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

      20. remove-double-neg 99.7%

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

   3. Simplified 99.7%

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

   5. Taylor expanded in y around 0 100.0%

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

      1. associate-*r/ 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around 0 87.8%

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

      1. metadata-eval 87.8%

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

      2. distribute-lft-neg-in 87.8%

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

      3. associate-*r/ 87.8%

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

      4. associate-*l/ 87.8%

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

      5. *-commutative 87.8%

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

      6. distribute-rgt-neg-in 87.8%

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

      7. distribute-neg-frac 87.8%

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

      8. metadata-eval 87.8%

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

   10. Simplified 87.8%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{+52} \lor \neg \left(x \leq 4.5 \cdot 10^{+155}\right):\\ \;\;\;\;4 + 4 \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;4 + z \cdot \frac{-4}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.4% 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 99.2%

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

   1. +-commutative 99.2%

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

   2. associate-*l/ 99.7%

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

   3. *-commutative 99.7%

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

   4. fma-def 99.7%

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

   5. +-commutative 99.7%

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

   6. associate--l+ 99.7%

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

   7. +-commutative 99.7%

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

   8. associate-+l- 99.7%

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

   9. sub-neg 99.7%

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

   10. remove-double-neg 99.7%

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

   11. distribute-neg-in 99.7%

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

   12. +-commutative 99.7%

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

   13. distribute-neg-in 99.7%

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

   14. distribute-lft-neg-out 99.7%

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

   15. sub-neg 99.7%

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

   16. distribute-lft-neg-out 99.7%

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

   17. distribute-rgt-neg-in 99.7%

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

   18. fma-neg 99.7%

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

   19. metadata-eval 99.7%

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

   20. remove-double-neg 99.7%

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

3. Simplified 99.7%

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

5. Taylor expanded in y around inf 32.5%

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

6. Final simplification 32.5%

   \[\leadsto 4 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024011 
(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)))