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

Percentage Accurate: 99.9% → 100.0%

Time: 6.7s

Alternatives: 8

Speedup: 1.3×

• 21.5% 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: 100.0% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 86.8

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

5. Applied rewrites 86.8%

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

6. Taylor expanded in z around 0

   \[\leadsto \color{blue}{\frac{-2 \cdot z + 4 \cdot \left(x - y\right)}{z}} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. div-add N/A

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

   3. associate-*r/ N/A

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

   4. *-commutative N/A

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

   5. associate-/l* N/A

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

   6. *-inverses N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower--.f64 100.0

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

8. Applied rewrites 100.0%

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

Alternative 2: 66.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{z} \cdot -4\\ t_1 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ t_2 := \frac{4 \cdot x}{z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+159}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -200000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -0.1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t\_1 \leq 10^{+106}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ y z) -4.0))
        (t_1 (/ (* 4.0 (- (- x y) (* z 0.5))) z))
        (t_2 (/ (* 4.0 x) z)))
   (if (<= t_1 -5e+159)
     t_0
     (if (<= t_1 -200000.0)
       t_2
       (if (<= t_1 -0.1) -2.0 (if (<= t_1 1e+106) t_0 t_2))))))

C

double code(double x, double y, double z) {
	double t_0 = (y / z) * -4.0;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double t_2 = (4.0 * x) / z;
	double tmp;
	if (t_1 <= -5e+159) {
		tmp = t_0;
	} else if (t_1 <= -200000.0) {
		tmp = t_2;
	} else if (t_1 <= -0.1) {
		tmp = -2.0;
	} else if (t_1 <= 1e+106) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_0 = (y / z) * (-4.0d0)
    t_1 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    t_2 = (4.0d0 * x) / z
    if (t_1 <= (-5d+159)) then
        tmp = t_0
    else if (t_1 <= (-200000.0d0)) then
        tmp = t_2
    else if (t_1 <= (-0.1d0)) then
        tmp = -2.0d0
    else if (t_1 <= 1d+106) then
        tmp = t_0
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y / z) * -4.0;
	double t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double t_2 = (4.0 * x) / z;
	double tmp;
	if (t_1 <= -5e+159) {
		tmp = t_0;
	} else if (t_1 <= -200000.0) {
		tmp = t_2;
	} else if (t_1 <= -0.1) {
		tmp = -2.0;
	} else if (t_1 <= 1e+106) {
		tmp = t_0;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y / z) * -4.0
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z
	t_2 = (4.0 * x) / z
	tmp = 0
	if t_1 <= -5e+159:
		tmp = t_0
	elif t_1 <= -200000.0:
		tmp = t_2
	elif t_1 <= -0.1:
		tmp = -2.0
	elif t_1 <= 1e+106:
		tmp = t_0
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y / z) * -4.0)
	t_1 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	t_2 = Float64(Float64(4.0 * x) / z)
	tmp = 0.0
	if (t_1 <= -5e+159)
		tmp = t_0;
	elseif (t_1 <= -200000.0)
		tmp = t_2;
	elseif (t_1 <= -0.1)
		tmp = -2.0;
	elseif (t_1 <= 1e+106)
		tmp = t_0;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y / z) * -4.0;
	t_1 = (4.0 * ((x - y) - (z * 0.5))) / z;
	t_2 = (4.0 * x) / z;
	tmp = 0.0;
	if (t_1 <= -5e+159)
		tmp = t_0;
	elseif (t_1 <= -200000.0)
		tmp = t_2;
	elseif (t_1 <= -0.1)
		tmp = -2.0;
	elseif (t_1 <= 1e+106)
		tmp = t_0;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y / z), $MachinePrecision] * -4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+159], t$95$0, If[LessEqual[t$95$1, -200000.0], t$95$2, If[LessEqual[t$95$1, -0.1], -2.0, If[LessEqual[t$95$1, 1e+106], t$95$0, t$95$2]]]]]]]

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) < -5.00000000000000003e159 or -0.10000000000000001 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 1.00000000000000009e106

   1. Initial program 98.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 69.1

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

   5. Applied rewrites 69.1%

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

   if -5.00000000000000003e159 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -2e5 or 1.00000000000000009e106 < (/.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 inf

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

      1. lower-*.f64 63.1

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

   5. Applied rewrites 63.1%

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

   if -2e5 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -0.10000000000000001

   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 93.2%

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

Alternative 3: 97.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+41} \lor \neg \left(t\_0 \leq 20000000000\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot 4}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -5e+41) (not (<= t_0 20000000000.0)))
     (/ (* (- x y) 4.0) z)
     (fma 4.0 (/ x 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 <= -5e+41) || !(t_0 <= 20000000000.0)) {
		tmp = ((x - y) * 4.0) / z;
	} else {
		tmp = fma(4.0, (x / z), -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if ((t_0 <= -5e+41) || !(t_0 <= 20000000000.0))
		tmp = Float64(Float64(Float64(x - y) * 4.0) / z);
	else
		tmp = fma(4.0, Float64(x / z), -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+41], N[Not[LessEqual[t$95$0, 20000000000.0]], $MachinePrecision]], N[(N[(N[(x - y), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision], N[(4.0 * N[(x / z), $MachinePrecision] + -2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -5.00000000000000022e41 or 2e10 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. metadata-eval N/A

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

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

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

      3. *-commutative N/A

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

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

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

      5. 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 99.4%

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

   if -5.00000000000000022e41 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 2e10

   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 95.6

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

   5. Applied rewrites 95.6%

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

   7. Applied rewrites 95.7%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -5 \cdot 10^{+41} \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq 20000000000\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot 4}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 65.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{+41} \lor \neg \left(t\_0 \leq -0.1\right):\\ \;\;\;\;\frac{y}{z} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* 4.0 (- (- x y) (* z 0.5))) z)))
   (if (or (<= t_0 -5e+41) (not (<= t_0 -0.1))) (* (/ y z) -4.0) -2.0)))

C

double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if ((t_0 <= -5e+41) || !(t_0 <= -0.1)) {
		tmp = (y / z) * -4.0;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
    if ((t_0 <= (-5d+41)) .or. (.not. (t_0 <= (-0.1d0)))) then
        tmp = (y / z) * (-4.0d0)
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	double tmp;
	if ((t_0 <= -5e+41) || !(t_0 <= -0.1)) {
		tmp = (y / z) * -4.0;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (4.0 * ((x - y) - (z * 0.5))) / z
	tmp = 0
	if (t_0 <= -5e+41) or not (t_0 <= -0.1):
		tmp = (y / z) * -4.0
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
	tmp = 0.0
	if ((t_0 <= -5e+41) || !(t_0 <= -0.1))
		tmp = Float64(Float64(y / z) * -4.0);
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (4.0 * ((x - y) - (z * 0.5))) / z;
	tmp = 0.0;
	if ((t_0 <= -5e+41) || ~((t_0 <= -0.1)))
		tmp = (y / z) * -4.0;
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[Or[LessEqual[t$95$0, -5e+41], N[Not[LessEqual[t$95$0, -0.1]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * -4.0), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -5.00000000000000022e41 or -0.10000000000000001 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 99.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 54.4

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

   5. Applied rewrites 54.4%

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

   if -5.00000000000000022e41 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < -0.10000000000000001

   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 91.5%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -5 \cdot 10^{+41} \lor \neg \left(\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \leq -0.1\right):\\ \;\;\;\;\frac{y}{z} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+61} \lor \neg \left(y \leq 3.7 \cdot 10^{+59}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.2e+61) (not (<= y 3.7e+59)))
   (fma (/ y z) -4.0 -2.0)
   (fma 4.0 (/ x z) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.2e+61) || !(y <= 3.7e+59)) {
		tmp = fma((y / z), -4.0, -2.0);
	} else {
		tmp = fma(4.0, (x / z), -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.2e+61) || !(y <= 3.7e+59))
		tmp = fma(Float64(y / z), -4.0, -2.0);
	else
		tmp = fma(4.0, Float64(x / z), -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.2e+61], N[Not[LessEqual[y, 3.7e+59]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * -4.0 + -2.0), $MachinePrecision], N[(4.0 * N[(x / z), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.19999999999999945e61 or 3.69999999999999997e59 < y

   1. Initial program 99.1%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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 90.5

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

   5. Applied rewrites 90.5%

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

   7. Applied rewrites 90.6%

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

   if -5.19999999999999945e61 < y < 3.69999999999999997e59

   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 87.3

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

   5. Applied rewrites 87.3%

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

   7. Applied rewrites 87.4%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+61} \lor \neg \left(y \leq 3.7 \cdot 10^{+59}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z}, -4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+61} \lor \neg \left(y \leq 3.7 \cdot 10^{+59}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -5.2e+61) (not (<= y 3.7e+59)))
   (fma (/ -4.0 z) y -2.0)
   (fma 4.0 (/ x z) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -5.2e+61) || !(y <= 3.7e+59)) {
		tmp = fma((-4.0 / z), y, -2.0);
	} else {
		tmp = fma(4.0, (x / z), -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -5.2e+61) || !(y <= 3.7e+59))
		tmp = fma(Float64(-4.0 / z), y, -2.0);
	else
		tmp = fma(4.0, Float64(x / z), -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -5.2e+61], N[Not[LessEqual[y, 3.7e+59]], $MachinePrecision]], N[(N[(-4.0 / z), $MachinePrecision] * y + -2.0), $MachinePrecision], N[(4.0 * N[(x / z), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.19999999999999945e61 or 3.69999999999999997e59 < y

   1. Initial program 99.1%

      \[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
   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 90.5

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

   5. Applied rewrites 90.5%

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

   if -5.19999999999999945e61 < y < 3.69999999999999997e59

   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 87.3

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

   5. Applied rewrites 87.3%

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

   7. Applied rewrites 87.4%

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

4. Final simplification 88.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+61} \lor \neg \left(y \leq 3.7 \cdot 10^{+59}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+124} \lor \neg \left(y \leq 5.2 \cdot 10^{+122}\right):\\ \;\;\;\;\frac{y}{z} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -2.8e+124) (not (<= y 5.2e+122)))
   (* (/ y z) -4.0)
   (fma 4.0 (/ x z) -2.0)))

C

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -2.8e+124) || !(y <= 5.2e+122)) {
		tmp = (y / z) * -4.0;
	} else {
		tmp = fma(4.0, (x / z), -2.0);
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if ((y <= -2.8e+124) || !(y <= 5.2e+122))
		tmp = Float64(Float64(y / z) * -4.0);
	else
		tmp = fma(4.0, Float64(x / z), -2.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := If[Or[LessEqual[y, -2.8e+124], N[Not[LessEqual[y, 5.2e+122]], $MachinePrecision]], N[(N[(y / z), $MachinePrecision] * -4.0), $MachinePrecision], N[(4.0 * N[(x / z), $MachinePrecision] + -2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.8e124 or 5.20000000000000015e122 < y

   1. Initial program 98.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.5

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

   5. Applied rewrites 80.5%

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

   if -2.8e124 < y < 5.20000000000000015e122

   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 81.9

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

   5. Applied rewrites 81.9%

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

   7. Applied rewrites 82.0%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{+124} \lor \neg \left(y \leq 5.2 \cdot 10^{+122}\right):\\ \;\;\;\;\frac{y}{z} \cdot -4\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(4, \frac{x}{z}, -2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.1% accurate, 28.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

3. Taylor expanded in z around inf

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

5. Applied rewrites 35.6%

   \[\leadsto \color{blue}{-2} \]
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 2024329 
(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))