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

Percentage Accurate: 99.9% → 99.9%

Time: 7.7s

Alternatives: 9

Speedup: 1.0×

• 20.5% 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, 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 100.0%

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

Alternative 2: 98.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -200000000000 \lor \neg \left(t\_0 \leq 1000000000000\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot 4}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -200000000000.0) (not (<= t_0 1000000000000.0)))
     (/ (* (- 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 <= -200000000000.0) || !(t_0 <= 1000000000000.0)) {
		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 <= -200000000000.0) || !(t_0 <= 1000000000000.0))
		tmp = Float64(Float64(Float64(x - y) * 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, -200000000000.0], N[Not[LessEqual[t$95$0, 1000000000000.0]], $MachinePrecision]], N[(N[(N[(x - y), $MachinePrecision] * 4.0), $MachinePrecision] / z), $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) < -2e11 or 1e12 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   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 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. metadata-eval N/A

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

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

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

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

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

      8. mul-1-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

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

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

      13. *-inverses N/A

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

      14. associate-/l* N/A

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

      15. associate-*l/ N/A

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

      16. distribute-neg-frac N/A

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

      17. mul-1-neg N/A

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

      18. *-commutative N/A

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

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

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

      20. mul-1-neg N/A

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

      21. lower-*.f64 N/A

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

   5. Applied rewrites 99.7%

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

   if -2e11 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 1e12

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

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

      2. div-add N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      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. lower-fma.f64 N/A

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

      10. lower-/.f64 95.6

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

   8. Applied rewrites 95.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -200000000000 \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq 1000000000000\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot 4}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -200000000000 \lor \neg \left(t\_0 \leq -1\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -200000000000.0) (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 <= -200000000000.0) || !(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 <= (-200000000000.0d0)) .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 <= -200000000000.0) || !(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 <= -200000000000.0) 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 <= -200000000000.0) || !(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 <= -200000000000.0) || ~((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, -200000000000.0], 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) < -2e11 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   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 inf

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

      1. lower-*.f64 61.0

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

   5. Applied rewrites 61.0%

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

   if -2e11 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 91.9%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -200000000000 \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -200000000000 \lor \neg \left(t\_0 \leq -1\right):\\ \;\;\;\;\frac{-4}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -200000000000.0) (not (<= t_0 -1.0)))
     (* (/ -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 <= -200000000000.0) || !(t_0 <= -1.0)) {
		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 <= (-200000000000.0d0)) .or. (.not. (t_0 <= (-1.0d0)))) 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 <= -200000000000.0) || !(t_0 <= -1.0)) {
		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 <= -200000000000.0) or not (t_0 <= -1.0):
		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 <= -200000000000.0) || !(t_0 <= -1.0))
		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 <= -200000000000.0) || ~((t_0 <= -1.0)))
		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, -200000000000.0], N[Not[LessEqual[t$95$0, -1.0]], $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) < -2e11 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   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 99.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 92.1%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 60.7%

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

   if -2e11 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 91.9%

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

4. Final simplification 72.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -200000000000 \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1\right):\\ \;\;\;\;\frac{-4}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+114} \lor \neg \left(x \leq 6.2 \cdot 10^{+49}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.4e+114) (not (<= x 6.2e+49)))
   (fma (/ x z) 4.0 -2.0)
   (fma -4.0 (/ y z) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.4e+114) || !(x <= 6.2e+49)) {
		tmp = fma((x / z), 4.0, -2.0);
	} else {
		tmp = fma(-4.0, (y / z), -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.4e+114) || !(x <= 6.2e+49))
		tmp = fma(Float64(x / z), 4.0, -2.0);
	else
		tmp = fma(-4.0, Float64(y / z), -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.4e+114], N[Not[LessEqual[x, 6.2e+49]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision], N[(-4.0 * N[(y / z), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e114 or 6.19999999999999985e49 < x

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 99.8%

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

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

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

      2. div-add N/A

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

      3. distribute-rgt-in N/A

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

      4. metadata-eval N/A

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

      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. lower-fma.f64 N/A

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

      10. lower-/.f64 84.2

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

   8. Applied rewrites 84.2%

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

   if -2.4e114 < x < 6.19999999999999985e49

   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. 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 92.0

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

   5. Applied rewrites 92.0%

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

   7. Applied rewrites 92.1%

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

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+114} \lor \neg \left(x \leq 6.2 \cdot 10^{+49}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+114} \lor \neg \left(x \leq 6.2 \cdot 10^{+49}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -2.4e+114) (not (<= x 6.2e+49)))
   (fma (/ 4.0 z) x -2.0)
   (fma -4.0 (/ y z) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -2.4e+114) || !(x <= 6.2e+49)) {
		tmp = fma((4.0 / z), x, -2.0);
	} else {
		tmp = fma(-4.0, (y / z), -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -2.4e+114) || !(x <= 6.2e+49))
		tmp = fma(Float64(4.0 / z), x, -2.0);
	else
		tmp = fma(-4.0, Float64(y / z), -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -2.4e+114], N[Not[LessEqual[x, 6.2e+49]], $MachinePrecision]], N[(N[(4.0 / z), $MachinePrecision] * x + -2.0), $MachinePrecision], N[(-4.0 * N[(y / z), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.4e114 or 6.19999999999999985e49 < x

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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. 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 84.0

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

   5. Applied rewrites 84.0%

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

   if -2.4e114 < x < 6.19999999999999985e49

   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. 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 92.0

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

   5. Applied rewrites 92.0%

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

   7. Applied rewrites 92.1%

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

4. Final simplification 89.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \cdot 10^{+114} \lor \neg \left(x \leq 6.2 \cdot 10^{+49}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+149} \lor \neg \left(x \leq 3.2 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{4 \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= x -3.3e+149) (not (<= x 3.2e+69)))
   (/ (* 4.0 x) z)
   (fma -4.0 (/ y z) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((x <= -3.3e+149) || !(x <= 3.2e+69)) {
		tmp = (4.0 * x) / z;
	} else {
		tmp = fma(-4.0, (y / z), -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((x <= -3.3e+149) || !(x <= 3.2e+69))
		tmp = Float64(Float64(4.0 * x) / z);
	else
		tmp = fma(-4.0, Float64(y / z), -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[x, -3.3e+149], N[Not[LessEqual[x, 3.2e+69]], $MachinePrecision]], N[(N[(4.0 * x), $MachinePrecision] / z), $MachinePrecision], N[(-4.0 * N[(y / z), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -3.3e149 or 3.19999999999999985e69 < x

   1. Initial program 99.9%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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 70.2

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

   5. Applied rewrites 70.2%

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

   if -3.3e149 < x < 3.19999999999999985e69

   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. 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 90.1

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

   5. Applied rewrites 90.1%

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

   7. Applied rewrites 90.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.3 \cdot 10^{+149} \lor \neg \left(x \leq 3.2 \cdot 10^{+69}\right):\\ \;\;\;\;\frac{4 \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-4, \frac{y}{z}, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 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 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 99.8%

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

Alternative 9: 33.9% 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 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 35.2%

   \[\leadsto \color{blue}{-2} \]
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 2024338 
(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))