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

Percentage Accurate: 99.9% → 100.0%

Time: 7.1s

Alternatives: 8

Speedup: 1.3×

• 21.1% 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: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.8

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

   8. lift--.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

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

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

   14. lower-fma.f64 N/A

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

   15. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

5. Taylor expanded in z around inf

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

   1. sub-neg N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower--.f64 100.0

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

7. Applied rewrites 100.0%

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

Alternative 2: 98.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+14} \lor \neg \left(t\_0 \leq 5000\right):\\ \;\;\;\;\frac{\left(y - x\right) \cdot -4}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -5e+14) (not (<= t_0 5000.0)))
     (/ (* (- 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+14) || !(t_0 <= 5000.0)) {
		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+14) || !(t_0 <= 5000.0))
		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+14], N[Not[LessEqual[t$95$0, 5000.0]], $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) < -5e14 or 5e3 < (/.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. 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 99.3

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

   5. Applied rewrites 99.3%

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

   if -5e14 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 5e3

   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 98.0%

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

   7. Applied rewrites 98.0%

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

4. Final simplification 98.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -5 \cdot 10^{+14} \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq 5000\right):\\ \;\;\;\;\frac{\left(y - x\right) \cdot -4}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -2000000 \lor \neg \left(t\_0 \leq -1\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -2000000.0) (not (<= t_0 -1.0))) (/ (* -4.0 y) z) -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 <= -2000000.0) || !(t_0 <= -1.0)) {
		tmp = (-4.0 * y) / z;
	} 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 <= (-2000000.0d0)) .or. (.not. (t_0 <= (-1.0d0)))) then
        tmp = ((-4.0d0) * y) / z
    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 <= -2000000.0) || !(t_0 <= -1.0)) {
		tmp = (-4.0 * y) / z;
	} 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 <= -2000000.0) or not (t_0 <= -1.0):
		tmp = (-4.0 * y) / z
	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 <= -2000000.0) || !(t_0 <= -1.0))
		tmp = Float64(Float64(-4.0 * y) / z);
	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 <= -2000000.0) || ~((t_0 <= -1.0)))
		tmp = (-4.0 * y) / z;
	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, -2000000.0], N[Not[LessEqual[t$95$0, -1.0]], $MachinePrecision]], N[(N[(-4.0 * y), $MachinePrecision] / z), $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) < -2e6 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 99.5%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 58.7

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

   5. Applied rewrites 58.7%

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

   if -2e6 < (/.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 98.1%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -2000000 \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+14} \lor \neg \left(t\_0 \leq -1\right):\\ \;\;\;\;\frac{x}{z} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -5e+14) (not (<= t_0 -1.0))) (* (/ x 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+14) || !(t_0 <= -1.0)) {
		tmp = (x / 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 <= (-5d+14)) .or. (.not. (t_0 <= (-1.0d0)))) then
        tmp = (x / 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 <= -5e+14) || !(t_0 <= -1.0)) {
		tmp = (x / 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 <= -5e+14) or not (t_0 <= -1.0):
		tmp = (x / 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 <= -5e+14) || !(t_0 <= -1.0))
		tmp = Float64(Float64(x / 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 <= -5e+14) || ~((t_0 <= -1.0)))
		tmp = (x / 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, -5e+14], N[Not[LessEqual[t$95$0, -1.0]], $MachinePrecision]], N[(N[(x / 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) < -5e14 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. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 46.3

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

   5. Applied rewrites 46.3%

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

   if -5e14 < (/.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 97.0%

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

4. Final simplification 61.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -5 \cdot 10^{+14} \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1\right):\\ \;\;\;\;\frac{x}{z} \cdot 4\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+74} \lor \neg \left(x \leq 2.1 \cdot 10^{+114}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.2e+74) (not (<= x 2.1e+114)))
   (fma (/ x z) 4.0 -2.0)
   (fma (/ y z) -4.0 -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.2e+74) || !(x <= 2.1e+114)) {
		tmp = fma((x / z), 4.0, -2.0);
	} else {
		tmp = fma((y / z), -4.0, -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.2e+74) || !(x <= 2.1e+114))
		tmp = fma(Float64(x / z), 4.0, -2.0);
	else
		tmp = fma(Float64(y / z), -4.0, -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.2e+74], N[Not[LessEqual[x, 2.1e+114]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * -4.0 + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000001e74 or 2.1e114 < x

   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 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 86.6

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 86.7%

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

   if -2.2000000000000001e74 < x < 2.1e114

   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 90.1%

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

   7. Applied rewrites 90.1%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+74} \lor \neg \left(x \leq 2.1 \cdot 10^{+114}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+85} \lor \neg \left(y \leq 1.35 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.9e+85) (not (<= y 1.35e-15)))
   (/ (* -4.0 y) z)
   (fma (/ x z) 4.0 -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.9e+85) || !(y <= 1.35e-15)) {
		tmp = (-4.0 * y) / z;
	} else {
		tmp = fma((x / z), 4.0, -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.9e+85) || !(y <= 1.35e-15))
		tmp = Float64(Float64(-4.0 * y) / z);
	else
		tmp = fma(Float64(x / z), 4.0, -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.9e+85], N[Not[LessEqual[y, 1.35e-15]], $MachinePrecision]], N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.89999999999999997e85 or 1.35000000000000005e-15 < y

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -2.89999999999999997e85 < y < 1.35000000000000005e-15

   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 89.0

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

   5. Applied rewrites 89.0%

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

   7. Applied rewrites 89.1%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+85} \lor \neg \left(y \leq 1.35 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+85} \lor \neg \left(y \leq 1.35 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.9e+85) (not (<= y 1.35e-15)))
   (/ (* -4.0 y) z)
   (fma (/ 4.0 z) x -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.9e+85) || !(y <= 1.35e-15)) {
		tmp = (-4.0 * y) / z;
	} else {
		tmp = fma((4.0 / z), x, -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.9e+85) || !(y <= 1.35e-15))
		tmp = Float64(Float64(-4.0 * y) / z);
	else
		tmp = fma(Float64(4.0 / z), x, -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.9e+85], N[Not[LessEqual[y, 1.35e-15]], $MachinePrecision]], N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision], N[(N[(4.0 / z), $MachinePrecision] * x + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.89999999999999997e85 or 1.35000000000000005e-15 < y

   1. Initial program 99.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 70.4

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

   5. Applied rewrites 70.4%

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

   if -2.89999999999999997e85 < y < 1.35000000000000005e-15

   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 89.0

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

   5. Applied rewrites 89.0%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.9 \cdot 10^{+85} \lor \neg \left(y \leq 1.35 \cdot 10^{-15}\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.6% 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 31.0%

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

Developer Target 1: 97.9% 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 2024324 
(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))