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

Percentage Accurate: 99.8% → 99.8%

Time: 6.9s

Alternatives: 8

Speedup: N/A×

• 20.9% 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: 99.8% 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: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

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 x around 0

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

5. Applied rewrites 99.8%

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

Alternative 2: 97.4% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -1e+14) (not (<= t_0 -2.0)))
     (* (- x y) (/ 4.0 z))
     (fma (/ 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 <= -1e+14) || !(t_0 <= -2.0)) {
		tmp = (x - y) * (4.0 / z);
	} else {
		tmp = fma((x / 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 <= -1e+14) || !(t_0 <= -2.0))
		tmp = Float64(Float64(x - y) * Float64(4.0 / z));
	else
		tmp = fma(Float64(x / 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, -1e+14], N[Not[LessEqual[t$95$0, -2.0]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] * N[(4.0 / z), $MachinePrecision]), $MachinePrecision], N[(N[(x / 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) < -1e14 or -2 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   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 z around 0

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. metadata-eval N/A

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

      5. associate-*r/ N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

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

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. *-inverses N/A

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

      14. associate-/l* N/A

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

      15. associate-*l/ N/A

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

      16. distribute-neg-frac N/A

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

      17. mul-1-neg N/A

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

      18. *-commutative N/A

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

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

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

      20. mul-1-neg N/A

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

      21. lower-*.f64 N/A

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

   5. Applied rewrites 99.1%

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

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

   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

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. div-add N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. associate-/l* N/A

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

      15. *-inverses N/A

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

      16. metadata-eval 100.0

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

   8. Applied rewrites 100.0%

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

4. Final simplification 99.3%

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

Alternative 3: 66.2% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -1e+14) (not (<= t_0 -2.0))) (* (/ 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 <= -1e+14) || !(t_0 <= -2.0)) {
		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 <= (-1d+14)) .or. (.not. (t_0 <= (-2.0d0)))) 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 <= -1e+14) || !(t_0 <= -2.0)) {
		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 <= -1e+14) or not (t_0 <= -2.0):
		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 <= -1e+14) || !(t_0 <= -2.0))
		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 <= -1e+14) || ~((t_0 <= -2.0)))
		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, -1e+14], N[Not[LessEqual[t$95$0, -2.0]], $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) < -1e14 or -2 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   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 inf

      \[\leadsto \color{blue}{-4 \cdot \frac{y}{z}} \]
   4. 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 57.1

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

   5. Applied rewrites 57.1%

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

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

   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 98.8%

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

4. Final simplification 69.2%

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

Alternative 4: 66.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -1e+14) (not (<= t_0 -2.0))) (* (/ -4.0 z) y) -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 <= -1e+14) || !(t_0 <= -2.0)) {
		tmp = (-4.0 / z) * y;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    if ((t_0 <= (-1d+14)) .or. (.not. (t_0 <= (-2.0d0)))) then
        tmp = ((-4.0d0) / z) * y
    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 <= -1e+14) || !(t_0 <= -2.0)) {
		tmp = (-4.0 / z) * y;
	} 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 <= -1e+14) or not (t_0 <= -2.0):
		tmp = (-4.0 / z) * y
	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 <= -1e+14) || !(t_0 <= -2.0))
		tmp = Float64(Float64(-4.0 / z) * y);
	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 <= -1e+14) || ~((t_0 <= -2.0)))
		tmp = (-4.0 / z) * y;
	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, -1e+14], N[Not[LessEqual[t$95$0, -2.0]], $MachinePrecision]], N[(N[(-4.0 / z), $MachinePrecision] * y), $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) < -1e14 or -2 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   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 inf

      \[\leadsto \color{blue}{-4 \cdot \frac{y}{z}} \]
   4. 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 57.1

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

   5. Applied rewrites 57.1%

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

   7. Applied rewrites 57.0%

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

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

   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 98.8%

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

4. Final simplification 69.0%

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

Alternative 5: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.9 \cdot 10^{-57} \lor \neg \left(y \leq 1.05 \cdot 10^{+86}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -9.9e-57) (not (<= y 1.05e+86)))
   (fma (/ -4.0 z) y -2.0)
   (fma (/ x z) 4.0 -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -9.9e-57) || !(y <= 1.05e+86)) {
		tmp = fma((-4.0 / z), y, -2.0);
	} else {
		tmp = fma((x / z), 4.0, -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -9.9e-57) || !(y <= 1.05e+86))
		tmp = fma(Float64(-4.0 / z), y, -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, -9.9e-57], N[Not[LessEqual[y, 1.05e+86]], $MachinePrecision]], N[(N[(-4.0 / z), $MachinePrecision] * y + -2.0), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.8999999999999995e-57 or 1.0499999999999999e86 < y

   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 x around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. div-add N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. metadata-eval 84.8

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

   5. Applied rewrites 84.8%

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

   if -9.8999999999999995e-57 < y < 1.0499999999999999e86

   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 x around 0

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. div-add N/A

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

      4. distribute-rgt-in N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

      10. *-inverses N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. associate-/l* N/A

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

      15. *-inverses N/A

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

      16. metadata-eval 90.5

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

   8. Applied rewrites 90.5%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.9 \cdot 10^{-57} \lor \neg \left(y \leq 1.05 \cdot 10^{+86}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.9 \cdot 10^{-57} \lor \neg \left(y \leq 1.05 \cdot 10^{+86}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -9.9e-57) (not (<= y 1.05e+86)))
   (fma (/ -4.0 z) y -2.0)
   (fma (/ 4.0 z) x -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -9.9e-57) || !(y <= 1.05e+86)) {
		tmp = fma((-4.0 / z), y, -2.0);
	} else {
		tmp = fma((4.0 / z), x, -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -9.9e-57) || !(y <= 1.05e+86))
		tmp = fma(Float64(-4.0 / z), y, -2.0);
	else
		tmp = fma(Float64(4.0 / z), x, -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -9.9e-57], N[Not[LessEqual[y, 1.05e+86]], $MachinePrecision]], N[(N[(-4.0 / z), $MachinePrecision] * y + -2.0), $MachinePrecision], N[(N[(4.0 / z), $MachinePrecision] * x + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.8999999999999995e-57 or 1.0499999999999999e86 < y

   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 x around 0

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

      1. associate-*r/ N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. div-add N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. metadata-eval 84.8

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

   5. Applied rewrites 84.8%

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

   if -9.8999999999999995e-57 < y < 1.0499999999999999e86

   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 0

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

      1. metadata-eval N/A

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

      2. fp-cancel-sign-sub-inv N/A

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

      3. div-add N/A

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

      4. distribute-lft-in N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. associate-*l* N/A

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

      8. associate-/l* N/A

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

      9. *-inverses N/A

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

      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. metadata-eval N/A

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

      13. *-inverses N/A

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

      14. associate-/l* N/A

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

      15. lower-fma.f64 N/A

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

      16. associate-*r/ N/A

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

      17. metadata-eval N/A

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

      18. lower-/.f64 N/A

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

      19. associate-/l* N/A

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

      20. *-inverses N/A

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

      21. metadata-eval 90.4

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

   5. Applied rewrites 90.4%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.9 \cdot 10^{-57} \lor \neg \left(y \leq 1.05 \cdot 10^{+86}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{+174} \lor \neg \left(x \leq 3.8 \cdot 10^{+123}\right):\\ \;\;\;\;\frac{4 \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -6.4e+174) (not (<= x 3.8e+123)))
   (/ (* 4.0 x) z)
   (fma (/ -4.0 z) y -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -6.4e+174) || !(x <= 3.8e+123)) {
		tmp = (4.0 * x) / z;
	} else {
		tmp = fma((-4.0 / z), y, -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -6.4e+174) || !(x <= 3.8e+123))
		tmp = Float64(Float64(4.0 * x) / z);
	else
		tmp = fma(Float64(-4.0 / z), y, -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -6.4e+174], N[Not[LessEqual[x, 3.8e+123]], $MachinePrecision]], N[(N[(4.0 * x), $MachinePrecision] / z), $MachinePrecision], N[(N[(-4.0 / z), $MachinePrecision] * y + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.4000000000000001e174 or 3.79999999999999994e123 < x

   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 inf

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

      1. lower-*.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if -6.4000000000000001e174 < x < 3.79999999999999994e123

   1. Initial program 99.0%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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. associate-*r/ N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. div-add N/A

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

      5. associate-*l/ N/A

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

      6. metadata-eval N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. *-commutative N/A

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

      10. associate-*r* N/A

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

      11. metadata-eval N/A

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

      12. lower-fma.f64 N/A

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

      13. metadata-eval N/A

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

      14. associate-*r/ N/A

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

      15. metadata-eval N/A

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

      16. lower-/.f64 N/A

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

      17. associate-/l* N/A

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

      18. *-inverses N/A

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

      19. metadata-eval 85.6

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

   5. Applied rewrites 85.6%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -6.4 \cdot 10^{+174} \lor \neg \left(x \leq 3.8 \cdot 10^{+123}\right):\\ \;\;\;\;\frac{4 \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.0% 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 99.2%

   \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
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: 98.1% 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 2024339 
(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))