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

Percentage Accurate: 99.8% → 99.8%

Time: 7.3s

Alternatives: 11

Speedup: N/A×

• 20.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

C

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * ((x - y) - (z * 0.5d0))) / z
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

Python

def code(x, y, z):
	return (4.0 * ((x - y) - (z * 0.5))) / z

Julia

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * ((x - y) - (z * 0.5))) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{4}{\frac{z}{\mathsf{fma}\left(-0.5, z, x - y\right)}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ 4.0 (/ z (fma -0.5 z (- x y)))))

C

double code(double x, double y, double z) {
	return 4.0 / (z / fma(-0.5, z, (x - y)));
}

Julia

function code(x, y, z)
	return Float64(4.0 / Float64(z / fma(-0.5, z, Float64(x - y))))
end

Wolfram

code[x_, y_, z_] := N[(4.0 / N[(z / N[(-0.5 * z + N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. clear-num N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 N/A

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

   7. lower-/.f64 99.8

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

   8. lift--.f64 N/A

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

   9. sub-neg N/A

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

   10. +-commutative N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

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

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

   14. lower-fma.f64 N/A

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

   15. metadata-eval 99.8

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

4. Applied rewrites 99.8%

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

Alternative 2: 66.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 4}{z}\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;t\_1 \leq 10^{+295}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{z} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x 4.0) z)) (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)))
   (if (<= t_1 -4e+35)
     t_0
     (if (<= t_1 -1.0) -2.0 (if (<= t_1 1e+295) t_0 (* (/ -4.0 z) y))))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / z;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -4e+35) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 1e+295) {
		tmp = t_0;
	} else {
		tmp = (-4.0 / z) * y;
	}
	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 = (x * 4.0d0) / z
    t_1 = (((x - y) - (0.5d0 * z)) * 4.0d0) / z
    if (t_1 <= (-4d+35)) then
        tmp = t_0
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else if (t_1 <= 1d+295) then
        tmp = t_0
    else
        tmp = ((-4.0d0) / z) * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / z;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -4e+35) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else if (t_1 <= 1e+295) {
		tmp = t_0;
	} else {
		tmp = (-4.0 / z) * y;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (x * 4.0) / z
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z
	tmp = 0
	if t_1 <= -4e+35:
		tmp = t_0
	elif t_1 <= -1.0:
		tmp = -2.0
	elif t_1 <= 1e+295:
		tmp = t_0
	else:
		tmp = (-4.0 / z) * y
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 4.0) / z)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	tmp = 0.0
	if (t_1 <= -4e+35)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 1e+295)
		tmp = t_0;
	else
		tmp = Float64(Float64(-4.0 / z) * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (x * 4.0) / z;
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	tmp = 0.0;
	if (t_1 <= -4e+35)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	elseif (t_1 <= 1e+295)
		tmp = t_0;
	else
		tmp = (-4.0 / z) * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 4.0), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+35], t$95$0, If[LessEqual[t$95$1, -1.0], -2.0, If[LessEqual[t$95$1, 1e+295], t$95$0, N[(N[(-4.0 / z), $MachinePrecision] * y), $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) < -3.9999999999999999e35 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z) < 9.9999999999999998e294

   1. Initial program 99.3%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 60.9

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

   5. Applied rewrites 60.9%

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

   if -3.9999999999999999e35 < (/.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 91.9%

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

   if 9.9999999999999998e294 < (/.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

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

      1. *-lft-identity N/A

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

      2. associate-*l/ N/A

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

      3. associate-*l* N/A

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

      4. metadata-eval N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. associate-*r/ N/A

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

      8. metadata-eval N/A

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

      9. distribute-neg-frac N/A

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

      10. metadata-eval N/A

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

      11. lower-/.f64 73.8

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

   5. Applied rewrites 73.8%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -4 \cdot 10^{+35}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq 10^{+295}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{z} \cdot 4\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+35}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (- x y) z) 4.0)) (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)))
   (if (<= t_1 -4e+35) t_0 (if (<= t_1 -1.0) (fma (/ -4.0 z) y -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((x - y) / z) * 4.0;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -4e+35) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = fma((-4.0 / z), y, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(x - y) / z) * 4.0)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	tmp = 0.0
	if (t_1 <= -4e+35)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = fma(Float64(-4.0 / z), y, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * 4.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+35], t$95$0, If[LessEqual[t$95$1, -1.0], N[(N[(-4.0 / z), $MachinePrecision] * y + -2.0), $MachinePrecision], t$95$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) < -3.9999999999999999e35 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -3.9999999999999999e35 < (/.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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-in N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

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

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

      19. *-lft-identity N/A

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

      20. associate-*l/ N/A

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

      21. associate-*l* N/A

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

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

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

      23. metadata-eval N/A

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

   7. Applied rewrites 98.6%

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

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -4 \cdot 10^{+35}:\\ \;\;\;\;\frac{x - y}{z} \cdot 4\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z} \cdot 4\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 66.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_0 \leq -2000000000:\\ \;\;\;\;\frac{y}{z} \cdot -4\\ \mathbf{elif}\;t\_0 \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)))
   (if (<= t_0 -2000000000.0)
     (* (/ y z) -4.0)
     (if (<= t_0 -1.0) -2.0 (/ (* -4.0 y) z)))))

C

double code(double x, double y, double z) {
	double t_0 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_0 <= -2000000000.0) {
		tmp = (y / z) * -4.0;
	} else if (t_0 <= -1.0) {
		tmp = -2.0;
	} 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) :: tmp
    t_0 = (((x - y) - (0.5d0 * z)) * 4.0d0) / z
    if (t_0 <= (-2000000000.0d0)) then
        tmp = (y / z) * (-4.0d0)
    else if (t_0 <= (-1.0d0)) then
        tmp = -2.0d0
    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 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_0 <= -2000000000.0) {
		tmp = (y / z) * -4.0;
	} else if (t_0 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = (-4.0 * y) / z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (((x - y) - (0.5 * z)) * 4.0) / z
	tmp = 0
	if t_0 <= -2000000000.0:
		tmp = (y / z) * -4.0
	elif t_0 <= -1.0:
		tmp = -2.0
	else:
		tmp = (-4.0 * y) / z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	tmp = 0.0
	if (t_0 <= -2000000000.0)
		tmp = Float64(Float64(y / z) * -4.0);
	elseif (t_0 <= -1.0)
		tmp = -2.0;
	else
		tmp = Float64(Float64(-4.0 * y) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (((x - y) - (0.5 * z)) * 4.0) / z;
	tmp = 0.0;
	if (t_0 <= -2000000000.0)
		tmp = (y / z) * -4.0;
	elseif (t_0 <= -1.0)
		tmp = -2.0;
	else
		tmp = (-4.0 * y) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, -2000000000.0], N[(N[(y / z), $MachinePrecision] * -4.0), $MachinePrecision], If[LessEqual[t$95$0, -1.0], -2.0, 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) < -2e9

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 98.7

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

   5. Applied rewrites 98.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 50.6%

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

   if -2e9 < (/.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 96.4%

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

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

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 45.7

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

   5. Applied rewrites 45.7%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -2000000000:\\ \;\;\;\;\frac{y}{z} \cdot -4\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{-4 \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 66.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y}{z} \cdot -4\\ t_1 := \frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z}\\ \mathbf{if}\;t\_1 \leq -2000000000:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_1 \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ y z) -4.0)) (t_1 (/ (* (- (- x y) (* 0.5 z)) 4.0) z)))
   (if (<= t_1 -2000000000.0) t_0 (if (<= t_1 -1.0) -2.0 t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y / z) * -4.0;
	double t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -2000000000.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = t_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) :: t_1
    real(8) :: tmp
    t_0 = (y / z) * (-4.0d0)
    t_1 = (((x - y) - (0.5d0 * z)) * 4.0d0) / z
    if (t_1 <= (-2000000000.0d0)) then
        tmp = t_0
    else if (t_1 <= (-1.0d0)) then
        tmp = -2.0d0
    else
        tmp = t_0
    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 = (((x - y) - (0.5 * z)) * 4.0) / z;
	double tmp;
	if (t_1 <= -2000000000.0) {
		tmp = t_0;
	} else if (t_1 <= -1.0) {
		tmp = -2.0;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y / z) * -4.0
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z
	tmp = 0
	if t_1 <= -2000000000.0:
		tmp = t_0
	elif t_1 <= -1.0:
		tmp = -2.0
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y / z) * -4.0)
	t_1 = Float64(Float64(Float64(Float64(x - y) - Float64(0.5 * z)) * 4.0) / z)
	tmp = 0.0
	if (t_1 <= -2000000000.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y / z) * -4.0;
	t_1 = (((x - y) - (0.5 * z)) * 4.0) / z;
	tmp = 0.0;
	if (t_1 <= -2000000000.0)
		tmp = t_0;
	elseif (t_1 <= -1.0)
		tmp = -2.0;
	else
		tmp = t_0;
	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[(N[(N[(x - y), $MachinePrecision] - N[(0.5 * z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$1, -2000000000.0], t$95$0, If[LessEqual[t$95$1, -1.0], -2.0, t$95$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) < -2e9 or -1 < (/.f64 (*.f64 #s(literal 4 binary64) (-.f64 (-.f64 x y) (*.f64 z #s(literal 1/2 binary64)))) z)

   1. Initial program 99.4%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 99.2

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 48.3%

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

   if -2e9 < (/.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 96.4%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -2000000000:\\ \;\;\;\;\frac{y}{z} \cdot -4\\ \mathbf{elif}\;\frac{\left(\left(x - y\right) - 0.5 \cdot z\right) \cdot 4}{z} \leq -1:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot -4\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-0.5 - \frac{y}{z}\right) \cdot 4\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- -0.5 (/ y z)) 4.0)))
   (if (<= y -1.8e-40) t_0 (if (<= y 2.9e-6) (fma (/ x z) 4.0 -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (-0.5 - (y / z)) * 4.0;
	double tmp;
	if (y <= -1.8e-40) {
		tmp = t_0;
	} else if (y <= 2.9e-6) {
		tmp = fma((x / z), 4.0, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-0.5 - Float64(y / z)) * 4.0)
	tmp = 0.0
	if (y <= -1.8e-40)
		tmp = t_0;
	elseif (y <= 2.9e-6)
		tmp = fma(Float64(x / z), 4.0, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-0.5 - N[(y / z), $MachinePrecision]), $MachinePrecision] * 4.0), $MachinePrecision]}, If[LessEqual[y, -1.8e-40], t$95$0, If[LessEqual[y, 2.9e-6], N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e-40 or 2.9000000000000002e-6 < y

   1. Initial program 99.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 81.2%

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

   if -1.8e-40 < y < 2.9000000000000002e-6

   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 95.2

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

   5. Applied rewrites 95.2%

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

   7. Applied rewrites 95.3%

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

Alternative 7: 86.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, 4, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ -4.0 z) y -2.0)))
   (if (<= y -1.8e-40) t_0 (if (<= y 2.9e-6) (fma (/ x z) 4.0 -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((-4.0 / z), y, -2.0);
	double tmp;
	if (y <= -1.8e-40) {
		tmp = t_0;
	} else if (y <= 2.9e-6) {
		tmp = fma((x / z), 4.0, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(-4.0 / z), y, -2.0)
	tmp = 0.0
	if (y <= -1.8e-40)
		tmp = t_0;
	elseif (y <= 2.9e-6)
		tmp = fma(Float64(x / z), 4.0, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-4.0 / z), $MachinePrecision] * y + -2.0), $MachinePrecision]}, If[LessEqual[y, -1.8e-40], t$95$0, If[LessEqual[y, 2.9e-6], N[(N[(x / z), $MachinePrecision] * 4.0 + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e-40 or 2.9000000000000002e-6 < y

   1. Initial program 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.8

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-in N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

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

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

      19. *-lft-identity N/A

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

      20. associate-*l/ N/A

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

      21. associate-*l* N/A

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

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

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

      23. metadata-eval N/A

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

   7. Applied rewrites 81.0%

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

   if -1.8e-40 < y < 2.9000000000000002e-6

   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 95.2

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

   5. Applied rewrites 95.2%

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

   7. Applied rewrites 95.3%

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

Alternative 8: 85.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(\frac{4}{z}, x, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ -4.0 z) y -2.0)))
   (if (<= y -1.8e-40) t_0 (if (<= y 2.9e-6) (fma (/ 4.0 z) x -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = fma((-4.0 / z), y, -2.0);
	double tmp;
	if (y <= -1.8e-40) {
		tmp = t_0;
	} else if (y <= 2.9e-6) {
		tmp = fma((4.0 / z), x, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = fma(Float64(-4.0 / z), y, -2.0)
	tmp = 0.0
	if (y <= -1.8e-40)
		tmp = t_0;
	elseif (y <= 2.9e-6)
		tmp = fma(Float64(4.0 / z), x, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-4.0 / z), $MachinePrecision] * y + -2.0), $MachinePrecision]}, If[LessEqual[y, -1.8e-40], t$95$0, If[LessEqual[y, 2.9e-6], N[(N[(4.0 / z), $MachinePrecision] * x + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.8e-40 or 2.9000000000000002e-6 < y

   1. Initial program 99.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.8

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-in N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

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

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

      19. *-lft-identity N/A

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

      20. associate-*l/ N/A

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

      21. associate-*l* N/A

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

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

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

      23. metadata-eval N/A

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

   7. Applied rewrites 81.0%

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

   if -1.8e-40 < y < 2.9000000000000002e-6

   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 95.2

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

   5. Applied rewrites 95.2%

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

Alternative 9: 80.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x \cdot 4}{z}\\ \mathbf{if}\;x \leq -2.6 \cdot 10^{+149}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x 4.0) z)))
   (if (<= x -2.6e+149) t_0 (if (<= x 4.6e+152) (fma (/ -4.0 z) y -2.0) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (x * 4.0) / z;
	double tmp;
	if (x <= -2.6e+149) {
		tmp = t_0;
	} else if (x <= 4.6e+152) {
		tmp = fma((-4.0 / z), y, -2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(x * 4.0) / z)
	tmp = 0.0
	if (x <= -2.6e+149)
		tmp = t_0;
	elseif (x <= 4.6e+152)
		tmp = fma(Float64(-4.0 / z), y, -2.0);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x * 4.0), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[x, -2.6e+149], t$95$0, If[LessEqual[x, 4.6e+152], N[(N[(-4.0 / z), $MachinePrecision] * y + -2.0), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.59999999999999979e149 or 4.5999999999999997e152 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. lower-*.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if -2.59999999999999979e149 < x < 4.5999999999999997e152

   1. Initial program 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 99.9

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

      8. lift--.f64 N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. lift-*.f64 N/A

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

      12. *-commutative N/A

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

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

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

      14. lower-fma.f64 N/A

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

      15. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. div-sub N/A

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. distribute-lft-in N/A

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

      5. associate-/l* N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. mul-1-neg N/A

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

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

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

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

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

      12. metadata-eval N/A

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

      13. metadata-eval N/A

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

      14. distribute-lft-in N/A

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

      15. +-commutative N/A

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

      16. distribute-lft-in N/A

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

      17. metadata-eval N/A

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

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

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

      19. *-lft-identity N/A

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

      20. associate-*l/ N/A

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

      21. associate-*l* N/A

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

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

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

      23. metadata-eval N/A

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

   7. Applied rewrites 82.5%

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

4. Final simplification 83.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+149}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{+152}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-4}{z}, y, -2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 4}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 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 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 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 11: 34.4% 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 36.2%

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

Developer Target 1: 98.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (4.0d0 * (x / z)) - (2.0d0 + (4.0d0 * (y / z)))
end function

Java

public static double code(double x, double y, double z) {
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
}

Python

def code(x, y, z):
	return (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)))

Julia

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

MATLAB

function tmp = code(x, y, z)
	tmp = (4.0 * (x / z)) - (2.0 + (4.0 * (y / z)));
end

Wolfram

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

Reproduce

herbie shell --seed 2024294 
(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))