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

Percentage Accurate: 99.8% → 100.0%

Time: 7.3s

Alternatives: 10

Speedup: 1.3×

• 20.3% 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: 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 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 40.9%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 100.0%

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

Alternative 2: 66.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot x}{z}\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+17}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+87}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 x) z)) (t_1 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (<= t_1 -1e+17)
     t_0
     (if (<= t_1 -1.0) -2.0 (if (<= t_1 5e+87) t_0 (/ (* -4.0 y) z))))))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * x) / z;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -1e+17) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 5e+87) {
		tmp = t_0;
	} else {
		tmp = (-4.0 * y) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (4.0d0 * x) / z
    t_1 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    if (t_1 <= (-1d+17)) then
        tmp = t_0
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else if (t_1 <= 5d+87) then
        tmp = t_0
    else
        tmp = ((-4.0d0) * y) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 * x) / z;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if (t_1 <= -1e+17) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 5e+87) {
		tmp = t_0;
	} else {
		tmp = (-4.0 * y) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * x) / z
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z
	tmp = 0
	if t_1 <= -1e+17:
		tmp = t_0
	elif t_1 <= -1.0:
		tmp = -2.0
	elif t_1 <= 5e+87:
		tmp = t_0
	else:
		tmp = (-4.0 * y) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * x) / z)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if (t_1 <= -1e+17)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 5e+87)
		tmp = t_0;
	else
		tmp = Float64(Float64(-4.0 * y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * x) / z;
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	tmp = 0.0;
	if (t_1 <= -1e+17)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 5e+87)
		tmp = t_0;
	else
		tmp = (-4.0 * y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * x), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+17], t$95$0, If[LessEqual[t$95$1, -1.0], -2.0, If[LessEqual[t$95$1, 5e+87], t$95$0, N[(N[(-4.0 * y), $MachinePrecision] / z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -1e17 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 4.9999999999999998e87

   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 57.1

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

   5. Applied rewrites 57.1%

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

   if -1e17 < (/.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 96.6%

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

   if 4.9999999999999998e87 < (/.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 57.3

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

   5. Applied rewrites 57.3%

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

Alternative 3: 98.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -400000000 \lor \neg \left(t\_0 \leq 5000000000\right):\\ \;\;\;\;\frac{\left(y - x\right) \cdot -4}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -400000000.0) (not (<= t_0 5000000000.0)))
     (/ (* (- y x) -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 <= -400000000.0) || !(t_0 <= 5000000000.0)) {
		tmp = ((y - x) * -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 <= -400000000.0) || !(t_0 <= 5000000000.0))
		tmp = Float64(Float64(Float64(y - x) * -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, -400000000.0], N[Not[LessEqual[t$95$0, 5000000000.0]], $MachinePrecision]], N[(N[(N[(y - x), $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) < -4e8 or 5e9 < (/.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. 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 99.6

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

   5. Applied rewrites 99.6%

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

   if -4e8 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 5e9

   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 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -400000000 \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq 5000000000\right):\\ \;\;\;\;\frac{\left(y - x\right) \cdot -4}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -400000000 \lor \neg \left(t\_0 \leq 5000000000\right):\\ \;\;\;\;\frac{-4}{z} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -400000000.0) (not (<= t_0 5000000000.0)))
     (* (/ -4.0 z) (- y x))
     (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 <= -400000000.0) || !(t_0 <= 5000000000.0)) {
		tmp = (-4.0 / z) * (y - x);
	} 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 <= -400000000.0) || !(t_0 <= 5000000000.0))
		tmp = Float64(Float64(-4.0 / z) * Float64(y - x));
	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, -400000000.0], N[Not[LessEqual[t$95$0, 5000000000.0]], $MachinePrecision]], N[(N[(-4.0 / z), $MachinePrecision] * N[(y - x), $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) < -4e8 or 5e9 < (/.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 inf

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

   5. Applied rewrites 2.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 100.0%

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

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{4 \cdot \frac{x - y}{z}} \]
   10. 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. *-commutative N/A

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

       7. sub-neg N/A

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

       8. mul-1-neg N/A

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

       9. distribute-lft-out N/A

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

       10. remove-double-neg N/A

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

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

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

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

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

       13. mul-1-neg N/A

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

       14. mul-1-neg N/A

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

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

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

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

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

       17. distribute-lft-out N/A

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

       18. *-rgt-identity N/A

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

   11. Applied rewrites 99.3%

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

   if -4e8 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 5e9

   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 98.8

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

   5. Applied rewrites 98.8%

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

   7. Applied rewrites 98.8%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -400000000 \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq 5000000000\right):\\ \;\;\;\;\frac{-4}{z} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -400000000 \lor \neg \left(t\_0 \leq -1\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -400000000.0) (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 <= -400000000.0) || !(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 <= (-400000000.0d0)) .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 <= -400000000.0) || !(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 <= -400000000.0) 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 <= -400000000.0) || !(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 <= -400000000.0) || ~((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, -400000000.0], 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) < -4e8 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   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 inf

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

      1. lower-*.f64 49.6

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

   5. Applied rewrites 49.6%

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

   if -4e8 < (/.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 98.3%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -400000000 \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -1\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+132} \lor \neg \left(y \leq 1.1 \cdot 10^{+42}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.3e+132) (not (<= y 1.1e+42)))
   (fma (/ y z) -4.0 -2.0)
   (fma (/ x z) 4.0 -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -1.3e+132) || !(y <= 1.1e+42)) {
		tmp = fma((y / z), -4.0, -2.0);
	} else {
		tmp = fma((x / z), 4.0, -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -1.3e+132) || !(y <= 1.1e+42))
		tmp = fma(Float64(y / z), -4.0, -2.0);
	else
		tmp = fma(Float64(x / z), 4.0, -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -1.3e+132], N[Not[LessEqual[y, 1.1e+42]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * -4.0 + -2.0), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.3e132 or 1.1000000000000001e42 < y

   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 21.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-4 \cdot \frac{y + \frac{1}{2} \cdot z}{z}} \]
   7. 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-lft-in N/A

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

      3. associate-*r* N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

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

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

      8. metadata-eval N/A

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

      9. associate-*r* N/A

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

      10. distribute-lft-in N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. associate-*r/ N/A

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

      14. div-sub N/A

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

      15. sub-neg N/A

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

      16. mul-1-neg N/A

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

      17. distribute-lft-in N/A

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

   8. Applied rewrites 90.6%

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

   if -1.3e132 < y < 1.1000000000000001e42

   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. 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 93.5

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

   5. Applied rewrites 93.5%

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

   7. Applied rewrites 93.6%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+132} \lor \neg \left(y \leq 1.1 \cdot 10^{+42}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+188} \lor \neg \left(y \leq 1.65 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5e+188) (not (<= y 1.65e+160)))
   (/ (* -4.0 y) z)
   (fma (/ x z) 4.0 -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5e+188) || !(y <= 1.65e+160)) {
		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 <= -5e+188) || !(y <= 1.65e+160))
		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, -5e+188], N[Not[LessEqual[y, 1.65e+160]], $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 < -5.0000000000000001e188 or 1.6499999999999999e160 < y

   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 84.8

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

   5. Applied rewrites 84.8%

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

   if -5.0000000000000001e188 < y < 1.6499999999999999e160

   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. 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 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+188} \lor \neg \left(y \leq 1.65 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+188} \lor \neg \left(y \leq 1.65 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5e+188) (not (<= y 1.65e+160)))
   (/ (* -4.0 y) z)
   (fma (/ 4.0 z) x -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5e+188) || !(y <= 1.65e+160)) {
		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 <= -5e+188) || !(y <= 1.65e+160))
		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, -5e+188], N[Not[LessEqual[y, 1.65e+160]], $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 < -5.0000000000000001e188 or 1.6499999999999999e160 < y

   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 84.8

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

   5. Applied rewrites 84.8%

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

   if -5.0000000000000001e188 < y < 1.6499999999999999e160

   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. 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 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5 \cdot 10^{+188} \lor \neg \left(y \leq 1.65 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

Alternative 10: 34.3% 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 40.9%

   \[\leadsto \color{blue}{-2} \]
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 2024318 
(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))