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

Percentage Accurate: 99.9% → 99.9%

Time: 6.8s

Alternatives: 9

Speedup: 1.0×

• 20.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

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 - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

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 - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\mathsf{fma}\left(-0.5, z, x \cdot \frac{x}{y + x}\right) - y \cdot \frac{y}{y + x}\right)}{z} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (* 4.0 (- (fma -0.5 z (* x (/ x (+ y x)))) (* y (/ y (+ y x))))) z))

C

double code(double x, double y, double z) {
	return (4.0 * (fma(-0.5, z, (x * (x / (y + x)))) - (y * (y / (y + x))))) / z;
}

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(fma(-0.5, z, Float64(x * Float64(x / Float64(y + x)))) - Float64(y * Float64(y / Float64(y + x))))) / z)
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(-0.5 * z + N[(x * N[(x / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y * N[(y / N[(y + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift--.f64 N/A

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

   5. flip-- N/A

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

   6. div-sub N/A

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

   7. associate-+r- N/A

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

   8. lower--.f64 N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. distribute-lft-neg-in N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. associate-/l* N/A

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

   15. lower-*.f64 N/A

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

   16. lower-/.f64 N/A

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

   17. +-commutative N/A

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

   18. lower-+.f64 N/A

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

   19. associate-/l* N/A

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

   20. lower-*.f64 N/A

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

   21. lower-/.f64 N/A

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

   22. +-commutative N/A

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

   23. lower-+.f64 99.6

       \[\leadsto \frac{4 \cdot \left(\mathsf{fma}\left(-0.5, z, x \cdot \frac{x}{y + x}\right) - y \cdot \frac{y}{\color{blue}{y + x}}\right)}{z} \]

4. Applied rewrites 99.6%

   \[\leadsto \frac{4 \cdot \color{blue}{\left(\mathsf{fma}\left(-0.5, z, x \cdot \frac{x}{y + x}\right) - y \cdot \frac{y}{y + x}\right)}}{z} \]
5. Add Preprocessing

Alternative 2: 66.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{z} \cdot -4\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_1 \leq -2000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+140}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{4 \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ y z) -4.0)) (t_1 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (<= t_1 -2000.0)
     t_0
     (if (<= t_1 -1.0) -2.0 (if (<= t_1 5e+140) t_0 (/ (* 4.0 x) z))))))

C

double code(double x, double y, double z) {
	double t_0 = (y / z) * -4.0;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -2000.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 5e+140) {
		tmp = t_0;
	} else {
		tmp = (4.0 * x) / z;
	}
	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 = (y / z) * (-4.0d0)
    t_1 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    if (t_1 <= (-2000.0d0)) then
        tmp = t_0
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else if (t_1 <= 5d+140) then
        tmp = t_0
    else
        tmp = (4.0d0 * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y / z) * -4.0;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -2000.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 5e+140) {
		tmp = t_0;
	} else {
		tmp = (4.0 * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y / z) * -4.0
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z
	tmp = 0
	if t_1 <= -2000.0:
		tmp = t_0
	elif t_1 <= -1.0:
		tmp = -2.0
	elif t_1 <= 5e+140:
		tmp = t_0
	else:
		tmp = (4.0 * x) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y / z) * -4.0)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if (t_1 <= -2000.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 5e+140)
		tmp = t_0;
	else
		tmp = Float64(Float64(4.0 * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y / z) * -4.0;
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	tmp = 0.0;
	if (t_1 <= -2000.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 5e+140)
		tmp = t_0;
	else
		tmp = (4.0 * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / z), $MachinePrecision] * -4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -2000.0], t$95$0, If[LessEqual[t$95$1, -1.0], -2.0, If[LessEqual[t$95$1, 5e+140], t$95$0, N[(N[(4.0 * x), $MachinePrecision] / z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -2e3 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 5.00000000000000008e140

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. div-sub N/A

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

      7. associate-+r- N/A

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

      8. lower--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. associate-/l* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 100.0

          \[\leadsto \frac{4 \cdot \left(\mathsf{fma}\left(-0.5, z, x \cdot \frac{x}{y + x}\right) - y \cdot \frac{y}{\color{blue}{y + x}}\right)}{z} \]

   4. Applied rewrites 100.0%

      \[\leadsto \frac{4 \cdot \color{blue}{\left(\mathsf{fma}\left(-0.5, z, x \cdot \frac{x}{y + x}\right) - y \cdot \frac{y}{y + x}\right)}}{z} \]

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 58.5

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

   7. Applied rewrites 58.5%

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

   if -2e3 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.4%

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

   if 5.00000000000000008e140 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 98.3%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 62.4

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

   5. Applied rewrites 62.4%

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

Alternative 3: 97.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+39} \lor \neg \left(t\_0 \leq 2000000000000\right):\\ \;\;\;\;\frac{\left(y - x\right) \cdot -4}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -5e+39) (not (<= t_0 2000000000000.0)))
     (/ (* (- y x) -4.0) z)
     (fma (/ y z) -4.0 -2.0))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if ((t_0 <= -5e+39) || !(t_0 <= 2000000000000.0)) {
		tmp = ((y - x) * -4.0) / z;
	} else {
		tmp = fma((y / z), -4.0, -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if ((t_0 <= -5e+39) || !(t_0 <= 2000000000000.0))
		tmp = Float64(Float64(Float64(y - x) * -4.0) / z);
	else
		tmp = fma(Float64(y / z), -4.0, -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+39], N[Not[LessEqual[t$95$0, 2000000000000.0]], $MachinePrecision]], N[(N[(N[(y - x), $MachinePrecision] * -4.0), $MachinePrecision] / z), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * -4.0 + -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -5.00000000000000015e39 or 2e12 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 99.3%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. metadata-eval N/A

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

      2. distribute-lft-neg-in N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-neg-in N/A

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

      5. mul-1-neg N/A

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

      6. *-rgt-identity N/A

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

      7. *-inverses N/A

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

      8. associate-/l* N/A

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

      9. associate-*l/ N/A

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

      10. associate-*r/ N/A

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

      11. *-commutative N/A

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

      12. lower-*.f64 N/A

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

      13. *-commutative N/A

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

      14. associate-*r/ N/A

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

      15. associate-*l/ N/A

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

      16. associate-/l* N/A

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

      17. *-inverses N/A

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

      18. *-rgt-identity N/A

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

      19. mul-1-neg N/A

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

      20. neg-sub0 N/A

          \[\leadsto \frac{\color{blue}{\left(0 - \left(x - y\right)\right)} \cdot -4}{z} \]

      21. associate-+l- N/A

          \[\leadsto \frac{\color{blue}{\left(\left(0 - x\right) + y\right)} \cdot -4}{z} \]

      22. neg-sub0 N/A

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

      23. mul-1-neg N/A

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

      24. +-commutative N/A

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

      25. mul-1-neg N/A

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

      26. sub-neg N/A

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

      27. lower--.f64 99.1

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

   5. Applied rewrites 99.1%

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

   if -5.00000000000000015e39 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 2e12

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y + \frac{1}{2} \cdot z}{z} \cdot -4} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot z + y}}{z} \cdot -4 \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\frac{1}{2} \cdot z + \color{blue}{y \cdot 1}}{z} \cdot -4 \]

      4. cancel-sign-sub N/A

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

      5. *-rgt-identity N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. sub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 99.4%

      \[\leadsto \color{blue}{\left(\frac{y}{z} - -0.5\right) \cdot -4} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.4%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -5 \cdot 10^{+39} \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq 2000000000000\right):\\ \;\;\;\;\frac{\left(y - x\right) \cdot -4}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -2000 \lor \neg \left(t\_0 \leq -1\right):\\ \;\;\;\;\frac{y}{z} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -2000.0) (not (<= t_0 -1.0))) (* (/ y z) -4.0) -2.0)))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if ((t_0 <= -2000.0) || !(t_0 <= -1.0)) {
		tmp = (y / z) * -4.0;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    if ((t_0 <= (-2000.0d0)) .or. (.not. (t_0 <= (-1.0d0)))) then
        tmp = (y / z) * (-4.0d0)
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if ((t_0 <= -2000.0) || !(t_0 <= -1.0)) {
		tmp = (y / z) * -4.0;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * ((x - y) - (z * 0.5))) / z
	tmp = 0
	if (t_0 <= -2000.0) or not (t_0 <= -1.0):
		tmp = (y / z) * -4.0
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if ((t_0 <= -2000.0) || !(t_0 <= -1.0))
		tmp = Float64(Float64(y / z) * -4.0);
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	tmp = 0.0;
	if ((t_0 <= -2000.0) || ~((t_0 <= -1.0)))
		tmp = (y / z) * -4.0;
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -2000.0], N[Not[LessEqual[t$95$0, -1.0]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * -4.0), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -2e3 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 99.4%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. div-sub N/A

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

      7. associate-+r- N/A

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

      8. lower--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. associate-/l* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 99.4

          \[\leadsto \frac{4 \cdot \left(\mathsf{fma}\left(-0.5, z, x \cdot \frac{x}{y + x}\right) - y \cdot \frac{y}{\color{blue}{y + x}}\right)}{z} \]

   4. Applied rewrites 99.4%

      \[\leadsto \frac{4 \cdot \color{blue}{\left(\mathsf{fma}\left(-0.5, z, x \cdot \frac{x}{y + x}\right) - y \cdot \frac{y}{y + x}\right)}}{z} \]

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 54.0

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

   7. Applied rewrites 54.0%

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

   if -2e3 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-2} \]
   4. Step-by-step derivation

   5. Applied rewrites 97.4%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -2000 \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1\right):\\ \;\;\;\;\frac{y}{z} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+42} \lor \neg \left(y \leq 1.22 \cdot 10^{+61}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.8e+42) (not (<= y 1.22e+61)))
   (fma (/ y z) -4.0 -2.0)
   (fma (/ x z) 4.0 -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.8e+42) || !(y <= 1.22e+61)) {
		tmp = fma((y / z), -4.0, -2.0);
	} else {
		tmp = fma((x / z), 4.0, -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.8e+42) || !(y <= 1.22e+61))
		tmp = fma(Float64(y / z), -4.0, -2.0);
	else
		tmp = fma(Float64(x / z), 4.0, -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -6.8e+42], N[Not[LessEqual[y, 1.22e+61]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * -4.0 + -2.0), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.7999999999999995e42 or 1.22e61 < y

   1. Initial program 98.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\frac{y + \frac{1}{2} \cdot z}{z} \cdot -4} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot z + y}}{z} \cdot -4 \]

      3. *-rgt-identity N/A

         \[\leadsto \frac{\frac{1}{2} \cdot z + \color{blue}{y \cdot 1}}{z} \cdot -4 \]

      4. cancel-sign-sub N/A

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

      5. *-rgt-identity N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. distribute-neg-frac N/A

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

      11. sub-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower-*.f64 N/A

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

   5. Applied rewrites 93.9%

      \[\leadsto \color{blue}{\left(\frac{y}{z} - -0.5\right) \cdot -4} \]
   6. Step-by-step derivation

   7. Applied rewrites 93.9%

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

   if -6.7999999999999995e42 < y < 1.22e61

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 90.8

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

   5. Applied rewrites 90.8%

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

   7. Applied rewrites 90.8%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{+42} \lor \neg \left(y \leq 1.22 \cdot 10^{+61}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+125} \lor \neg \left(y \leq 3.2 \cdot 10^{+119}\right):\\ \;\;\;\;\frac{y}{z} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.65e+125) (not (<= y 3.2e+119)))
   (* (/ y z) -4.0)
   (fma (/ x z) 4.0 -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.65e+125) || !(y <= 3.2e+119)) {
		tmp = (y / z) * -4.0;
	} else {
		tmp = fma((x / z), 4.0, -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.65e+125) || !(y <= 3.2e+119))
		tmp = Float64(Float64(y / z) * -4.0);
	else
		tmp = fma(Float64(x / z), 4.0, -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.65e+125], N[Not[LessEqual[y, 3.2e+119]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65000000000000003e125 or 3.19999999999999989e119 < y

   1. Initial program 98.6%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. div-sub N/A

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

      7. associate-+r- N/A

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

      8. lower--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. associate-/l* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 98.6

          \[\leadsto \frac{4 \cdot \left(\mathsf{fma}\left(-0.5, z, x \cdot \frac{x}{y + x}\right) - y \cdot \frac{y}{\color{blue}{y + x}}\right)}{z} \]

   4. Applied rewrites 98.6%

      \[\leadsto \frac{4 \cdot \color{blue}{\left(\mathsf{fma}\left(-0.5, z, x \cdot \frac{x}{y + x}\right) - y \cdot \frac{y}{y + x}\right)}}{z} \]

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 82.5

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

   7. Applied rewrites 82.5%

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

   if -1.65000000000000003e125 < y < 3.19999999999999989e119

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 87.5

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

   5. Applied rewrites 87.5%

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

   7. Applied rewrites 87.6%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+125} \lor \neg \left(y \leq 3.2 \cdot 10^{+119}\right):\\ \;\;\;\;\frac{y}{z} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+125} \lor \neg \left(y \leq 3.2 \cdot 10^{+119}\right):\\ \;\;\;\;\frac{y}{z} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.65e+125) (not (<= y 3.2e+119)))
   (* (/ y z) -4.0)
   (fma (/ 4.0 z) x -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.65e+125) || !(y <= 3.2e+119)) {
		tmp = (y / z) * -4.0;
	} else {
		tmp = fma((4.0 / z), x, -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.65e+125) || !(y <= 3.2e+119))
		tmp = Float64(Float64(y / z) * -4.0);
	else
		tmp = fma(Float64(4.0 / z), x, -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.65e+125], N[Not[LessEqual[y, 3.2e+119]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * -4.0), $MachinePrecision], N[(N[(4.0 / z), $MachinePrecision] * x + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65000000000000003e125 or 3.19999999999999989e119 < y

   1. Initial program 98.6%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift--.f64 N/A

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

      5. flip-- N/A

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

      6. div-sub N/A

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

      7. associate-+r- N/A

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

      8. lower--.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. *-commutative N/A

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

      11. distribute-lft-neg-in N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. associate-/l* N/A

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

      15. lower-*.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. +-commutative N/A

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

      18. lower-+.f64 N/A

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

      19. associate-/l* N/A

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

      20. lower-*.f64 N/A

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

      21. lower-/.f64 N/A

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

      22. +-commutative N/A

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

      23. lower-+.f64 98.6

          \[\leadsto \frac{4 \cdot \left(\mathsf{fma}\left(-0.5, z, x \cdot \frac{x}{y + x}\right) - y \cdot \frac{y}{\color{blue}{y + x}}\right)}{z} \]

   4. Applied rewrites 98.6%

      \[\leadsto \frac{4 \cdot \color{blue}{\left(\mathsf{fma}\left(-0.5, z, x \cdot \frac{x}{y + x}\right) - y \cdot \frac{y}{y + x}\right)}}{z} \]

   5. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 82.5

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

   7. Applied rewrites 82.5%

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

   if -1.65000000000000003e125 < y < 3.19999999999999989e119

   1. Initial program 100.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - \frac{1}{2} \cdot z}{z}} \]
   4. Step-by-step derivation

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. distribute-lft-in N/A

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

      4. *-lft-identity N/A

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

      5. associate-*l/ N/A

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

      6. associate-*l* N/A

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

      7. associate-/l* N/A

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

      8. *-inverses N/A

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

      9. metadata-eval N/A

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

      10. metadata-eval N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. associate-*r/ N/A

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

      14. metadata-eval N/A

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

      15. lower-/.f64 87.5

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

   5. Applied rewrites 87.5%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+125} \lor \neg \left(y \leq 3.2 \cdot 10^{+119}\right):\\ \;\;\;\;\frac{y}{z} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

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 - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 9: 34.5% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := -2.0

Derivation

1. Initial program 99.6%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{-2} \]
4. Step-by-step derivation

5. Applied rewrites 39.7%

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

Developer Target 1: 98.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

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)) - (2.0d0 + (4.0d0 * (y / z)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024313 
(FPCore (x y z)
  :name "Data.Array.Repa.Algorithms.ColorRamp:rampColorHotToCold from repa-algorithms-3.4.0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (* 4 (/ x z)) (+ 2 (* 4 (/ y z)))))

  (/ (* 4.0 (- (- x y) (* z 0.5))) z))