Rosa's Benchmark

Percentage Accurate: 99.8% → 99.8%

Time: 28.9s

Alternatives: 6

Speedup: 1.2×

• 33.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))

C

double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.954929658551372d0 * x) - (0.12900613773279798d0 * ((x * x) * x))
end function

Java

public static double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Python

def code(x):
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))

Julia

function code(x)
	return Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(Float64(x * x) * x)))
end

MATLAB

function tmp = code(x)
	tmp = (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
end

Wolfram

code[x_] := N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))

C

double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.954929658551372d0 * x) - (0.12900613773279798d0 * ((x * x) * x))
end function

Java

public static double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Python

def code(x):
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))

Julia

function code(x)
	return Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(Float64(x * x) * x)))
end

MATLAB

function tmp = code(x)
	tmp = (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
end

Wolfram

code[x_] := N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(0.954929658551372 \cdot x\_m - 0.12900613773279798 \cdot \left(\sqrt{x\_m} \cdot {x\_m}^{2.5}\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (-
   (* 0.954929658551372 x_m)
   (* 0.12900613773279798 (* (sqrt x_m) (pow x_m 2.5))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((0.954929658551372 * x_m) - (0.12900613773279798 * (sqrt(x_m) * pow(x_m, 2.5))));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((0.954929658551372d0 * x_m) - (0.12900613773279798d0 * (sqrt(x_m) * (x_m ** 2.5d0))))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * ((0.954929658551372 * x_m) - (0.12900613773279798 * (Math.sqrt(x_m) * Math.pow(x_m, 2.5))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((0.954929658551372 * x_m) - (0.12900613773279798 * (math.sqrt(x_m) * math.pow(x_m, 2.5))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(0.954929658551372 * x_m) - Float64(0.12900613773279798 * Float64(sqrt(x_m) * (x_m ^ 2.5)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((0.954929658551372 * x_m) - (0.12900613773279798 * (sqrt(x_m) * (x_m ^ 2.5))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(0.954929658551372 * x$95$m), $MachinePrecision] - N[(0.12900613773279798 * N[(N[Sqrt[x$95$m], $MachinePrecision] * N[Power[x$95$m, 2.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. pow3 99.8%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \color{blue}{{x}^{3}} \]

   2. sqr-pow 50.7%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \color{blue}{\left({x}^{\left(\frac{3}{2}\right)} \cdot {x}^{\left(\frac{3}{2}\right)}\right)} \]

   3. pow2 50.7%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \color{blue}{{\left({x}^{\left(\frac{3}{2}\right)}\right)}^{2}} \]

   4. metadata-eval 50.7%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot {\left({x}^{\color{blue}{1.5}}\right)}^{2} \]

4. Applied egg-rr 50.7%

   \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \color{blue}{{\left({x}^{1.5}\right)}^{2}} \]
5. Step-by-step derivation

   1. unpow2 50.7%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \color{blue}{\left({x}^{1.5} \cdot {x}^{1.5}\right)} \]

   2. add-cbrt-cube 48.1%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left({x}^{1.5} \cdot \color{blue}{\sqrt[3]{\left({x}^{1.5} \cdot {x}^{1.5}\right) \cdot {x}^{1.5}}}\right) \]

   3. unpow2 48.1%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left({x}^{1.5} \cdot \sqrt[3]{\color{blue}{{\left({x}^{1.5}\right)}^{2}} \cdot {x}^{1.5}}\right) \]

   4. cbrt-prod 50.6%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left({x}^{1.5} \cdot \color{blue}{\left(\sqrt[3]{{\left({x}^{1.5}\right)}^{2}} \cdot \sqrt[3]{{x}^{1.5}}\right)}\right) \]

   5. pow-pow 50.6%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left({x}^{1.5} \cdot \left(\sqrt[3]{\color{blue}{{x}^{\left(1.5 \cdot 2\right)}}} \cdot \sqrt[3]{{x}^{1.5}}\right)\right) \]

   6. metadata-eval 50.6%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left({x}^{1.5} \cdot \left(\sqrt[3]{{x}^{\color{blue}{3}}} \cdot \sqrt[3]{{x}^{1.5}}\right)\right) \]

   7. rem-cbrt-cube 50.6%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left({x}^{1.5} \cdot \left(\color{blue}{x} \cdot \sqrt[3]{{x}^{1.5}}\right)\right) \]

   8. associate-*r* 50.6%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \color{blue}{\left(\left({x}^{1.5} \cdot x\right) \cdot \sqrt[3]{{x}^{1.5}}\right)} \]

   9. metadata-eval 50.6%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left({x}^{1.5} \cdot x\right) \cdot \sqrt[3]{{x}^{\color{blue}{\left(\frac{3}{2}\right)}}}\right) \]

   10. sqrt-pow1 50.6%

       \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left({x}^{1.5} \cdot x\right) \cdot \sqrt[3]{\color{blue}{\sqrt{{x}^{3}}}}\right) \]

   11. pow3 50.6%

       \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left({x}^{1.5} \cdot x\right) \cdot \sqrt[3]{\sqrt{\color{blue}{\left(x \cdot x\right) \cdot x}}}\right) \]

   12. pow2 50.6%

       \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left({x}^{1.5} \cdot x\right) \cdot \sqrt[3]{\sqrt{\color{blue}{{x}^{2}} \cdot x}}\right) \]

   13. sqrt-prod 50.6%

       \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left({x}^{1.5} \cdot x\right) \cdot \sqrt[3]{\color{blue}{\sqrt{{x}^{2}} \cdot \sqrt{x}}}\right) \]

   14. pow2 50.6%

       \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left({x}^{1.5} \cdot x\right) \cdot \sqrt[3]{\sqrt{\color{blue}{x \cdot x}} \cdot \sqrt{x}}\right) \]

   15. sqrt-prod 50.6%

       \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left({x}^{1.5} \cdot x\right) \cdot \sqrt[3]{\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \sqrt{x}}\right) \]

   16. add-cbrt-cube 50.7%

       \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left({x}^{1.5} \cdot x\right) \cdot \color{blue}{\sqrt{x}}\right) \]

6. Applied egg-rr 50.7%

   \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \color{blue}{\left(\left({x}^{1.5} \cdot x\right) \cdot \sqrt{x}\right)} \]
7. Step-by-step derivation

   1. *-commutative 50.7%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \color{blue}{\left(\sqrt{x} \cdot \left({x}^{1.5} \cdot x\right)\right)} \]

   2. pow-plus 50.7%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\sqrt{x} \cdot \color{blue}{{x}^{\left(1.5 + 1\right)}}\right) \]

   3. metadata-eval 50.7%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\sqrt{x} \cdot {x}^{\color{blue}{2.5}}\right) \]

8. Simplified 50.7%

   \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \color{blue}{\left(\sqrt{x} \cdot {x}^{2.5}\right)} \]
9. Add Preprocessing

Alternative 2: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \mathsf{fma}\left(0.954929658551372, x\_m, {x\_m}^{3} \cdot -0.12900613773279798\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (fma 0.954929658551372 x_m (* (pow x_m 3.0) -0.12900613773279798))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * fma(0.954929658551372, x_m, (pow(x_m, 3.0) * -0.12900613773279798));
}

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * fma(0.954929658551372, x_m, Float64((x_m ^ 3.0) * -0.12900613773279798)))
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(0.954929658551372 * x$95$m + N[(N[Power[x$95$m, 3.0], $MachinePrecision] * -0.12900613773279798), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
2. Step-by-step derivation

   1. fma-neg 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, -0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)} \]

   2. *-commutative 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, -\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.12900613773279798}\right) \]

   3. distribute-rgt-neg-in 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(-0.12900613773279798\right)}\right) \]

   4. unpow3 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{{x}^{3}} \cdot \left(-0.12900613773279798\right)\right) \]

   5. metadata-eval 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot \color{blue}{-0.12900613773279798}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot -0.12900613773279798\right)} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(0.954929658551372 \cdot x\_m - 0.12900613773279798 \cdot \left(x\_m \cdot \left(x\_m \cdot x\_m\right)\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (- (* 0.954929658551372 x_m) (* 0.12900613773279798 (* x_m (* x_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * ((0.954929658551372 * x_m) - (0.12900613773279798 * (x_m * (x_m * x_m))));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * ((0.954929658551372d0 * x_m) - (0.12900613773279798d0 * (x_m * (x_m * x_m))))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * ((0.954929658551372 * x_m) - (0.12900613773279798 * (x_m * (x_m * x_m))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * ((0.954929658551372 * x_m) - (0.12900613773279798 * (x_m * (x_m * x_m))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(Float64(0.954929658551372 * x_m) - Float64(0.12900613773279798 * Float64(x_m * Float64(x_m * x_m)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * ((0.954929658551372 * x_m) - (0.12900613773279798 * (x_m * (x_m * x_m))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(N[(0.954929658551372 * x$95$m), $MachinePrecision] - N[(0.12900613773279798 * N[(x$95$m * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
2. Add Preprocessing

3. Final simplification 99.8%

   \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right) \]
4. Add Preprocessing

Alternative 4: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x\_m \cdot x\_m\right)\right)\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (* x_s (* x_m (- 0.954929658551372 (* 0.12900613773279798 (* x_m x_m))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (x_m * (0.954929658551372 - (0.12900613773279798 * (x_m * x_m))));
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (x_m * (0.954929658551372d0 - (0.12900613773279798d0 * (x_m * x_m))))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (x_m * (0.954929658551372 - (0.12900613773279798 * (x_m * x_m))));
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (x_m * (0.954929658551372 - (0.12900613773279798 * (x_m * x_m))))

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m * Float64(0.954929658551372 - Float64(0.12900613773279798 * Float64(x_m * x_m)))))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (x_m * (0.954929658551372 - (0.12900613773279798 * (x_m * x_m))));
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(x$95$m * N[(0.954929658551372 - N[(0.12900613773279798 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
2. Step-by-step derivation

   1. fma-neg 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, -0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)} \]

   2. *-commutative 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, -\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.12900613773279798}\right) \]

   3. distribute-rgt-neg-in 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(-0.12900613773279798\right)}\right) \]

   4. unpow3 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{{x}^{3}} \cdot \left(-0.12900613773279798\right)\right) \]

   5. metadata-eval 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot \color{blue}{-0.12900613773279798}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot -0.12900613773279798\right)} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. fma-undefine 99.8%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x + {x}^{3} \cdot -0.12900613773279798} \]

   2. pow3 99.8%

      \[\leadsto 0.954929658551372 \cdot x + \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} \cdot -0.12900613773279798 \]

   3. *-commutative 99.8%

      \[\leadsto 0.954929658551372 \cdot x + \color{blue}{-0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

   4. metadata-eval 99.8%

      \[\leadsto 0.954929658551372 \cdot x + \color{blue}{\left(-0.12900613773279798\right)} \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

   5. cancel-sign-sub-inv 99.8%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

   6. associate-*r* 99.8%

      \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\left(0.12900613773279798 \cdot \left(x \cdot x\right)\right) \cdot x} \]

   7. distribute-rgt-out-- 99.8%

      \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

   8. *-commutative 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{\left(x \cdot x\right) \cdot 0.12900613773279798}\right) \]

   9. pow2 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{{x}^{2}} \cdot 0.12900613773279798\right) \]

6. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - {x}^{2} \cdot 0.12900613773279798\right)} \]
7. Step-by-step derivation

   1. pow2 99.8%

      \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{\left(x \cdot x\right)} \cdot 0.12900613773279798\right) \]

8. Applied egg-rr 99.8%

   \[\leadsto x \cdot \left(0.954929658551372 - \color{blue}{\left(x \cdot x\right)} \cdot 0.12900613773279798\right) \]

9. Final simplification 99.8%

   \[\leadsto x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot x\right)\right) \]
10. Add Preprocessing

Alternative 5: 52.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.75:\\ \;\;\;\;0.954929658551372 \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot -0.954929658551372\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2.75) (* 0.954929658551372 x_m) (* x_m -0.954929658551372))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.75) {
		tmp = 0.954929658551372 * x_m;
	} else {
		tmp = x_m * -0.954929658551372;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.75d0) then
        tmp = 0.954929658551372d0 * x_m
    else
        tmp = x_m * (-0.954929658551372d0)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	double tmp;
	if (x_m <= 2.75) {
		tmp = 0.954929658551372 * x_m;
	} else {
		tmp = x_m * -0.954929658551372;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	tmp = 0
	if x_m <= 2.75:
		tmp = 0.954929658551372 * x_m
	else:
		tmp = x_m * -0.954929658551372
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	tmp = 0.0
	if (x_m <= 2.75)
		tmp = Float64(0.954929658551372 * x_m);
	else
		tmp = Float64(x_m * -0.954929658551372);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m)
	tmp = 0.0;
	if (x_m <= 2.75)
		tmp = 0.954929658551372 * x_m;
	else
		tmp = x_m * -0.954929658551372;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * If[LessEqual[x$95$m, 2.75], N[(0.954929658551372 * x$95$m), $MachinePrecision], N[(x$95$m * -0.954929658551372), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2.75

   1. Initial program 99.8%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Step-by-step derivation

      1. fma-neg 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, -0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)} \]

      2. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(0.954929658551372, x, -\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.12900613773279798}\right) \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(-0.12900613773279798\right)}\right) \]

      4. unpow3 99.8%

         \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{{x}^{3}} \cdot \left(-0.12900613773279798\right)\right) \]

      5. metadata-eval 99.8%

         \[\leadsto \mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot \color{blue}{-0.12900613773279798}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot -0.12900613773279798\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 68.8%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 68.8%

         \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]

   7. Simplified 68.8%

      \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]

   if 2.75 < x

   1. Initial program 99.8%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Step-by-step derivation

      1. fma-neg 99.8%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, -0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)} \]

      2. *-commutative 99.8%

         \[\leadsto \mathsf{fma}\left(0.954929658551372, x, -\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.12900613773279798}\right) \]

      3. distribute-rgt-neg-in 99.8%

         \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(-0.12900613773279798\right)}\right) \]

      4. unpow3 99.8%

         \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{{x}^{3}} \cdot \left(-0.12900613773279798\right)\right) \]

      5. metadata-eval 99.8%

         \[\leadsto \mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot \color{blue}{-0.12900613773279798}\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot -0.12900613773279798\right)} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 0.4%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
   6. Step-by-step derivation

      1. *-commutative 0.4%

         \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]

   7. Simplified 0.4%

      \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]
   8. Step-by-step derivation

      1. *-commutative 0.4%

         \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]

      2. add-sqr-sqrt 0.4%

         \[\leadsto \color{blue}{\sqrt{0.954929658551372 \cdot x} \cdot \sqrt{0.954929658551372 \cdot x}} \]

      3. sqrt-unprod 0.3%

         \[\leadsto \color{blue}{\sqrt{\left(0.954929658551372 \cdot x\right) \cdot \left(0.954929658551372 \cdot x\right)}} \]

      4. *-commutative 0.3%

         \[\leadsto \sqrt{\left(0.954929658551372 \cdot x\right) \cdot \color{blue}{\left(x \cdot 0.954929658551372\right)}} \]

      5. *-commutative 0.3%

         \[\leadsto \sqrt{\color{blue}{\left(x \cdot 0.954929658551372\right)} \cdot \left(x \cdot 0.954929658551372\right)} \]

      6. swap-sqr 0.3%

         \[\leadsto \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(0.954929658551372 \cdot 0.954929658551372\right)}} \]

      7. pow2 0.3%

         \[\leadsto \sqrt{\color{blue}{{x}^{2}} \cdot \left(0.954929658551372 \cdot 0.954929658551372\right)} \]

      8. metadata-eval 0.3%

         \[\leadsto \sqrt{{x}^{2} \cdot \color{blue}{0.9118906527810399}} \]

   9. Applied egg-rr 0.3%

      \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot 0.9118906527810399}} \]

   10. Taylor expanded in x around -inf 6.3%

       \[\leadsto \color{blue}{-0.954929658551372 \cdot x} \]
   11. Step-by-step derivation

       1. *-commutative 6.3%

          \[\leadsto \color{blue}{x \cdot -0.954929658551372} \]

   12. Simplified 6.3%

       \[\leadsto \color{blue}{x \cdot -0.954929658551372} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.75:\\ \;\;\;\;0.954929658551372 \cdot x\\ \mathbf{else}:\\ \;\;\;\;x \cdot -0.954929658551372\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 5.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \left(x\_m \cdot -0.954929658551372\right) \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s (* x_m -0.954929658551372)))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * (x_m * -0.954929658551372);
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    code = x_s * (x_m * (-0.954929658551372d0))
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m) {
	return x_s * (x_m * -0.954929658551372);
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * (x_m * -0.954929658551372)

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * Float64(x_m * -0.954929658551372))
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * (x_m * -0.954929658551372);
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_] := N[(x$95$s * N[(x$95$m * -0.954929658551372), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
2. Step-by-step derivation

   1. fma-neg 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, -0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right)} \]

   2. *-commutative 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, -\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.12900613773279798}\right) \]

   3. distribute-rgt-neg-in 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(-0.12900613773279798\right)}\right) \]

   4. unpow3 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, \color{blue}{{x}^{3}} \cdot \left(-0.12900613773279798\right)\right) \]

   5. metadata-eval 99.8%

      \[\leadsto \mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot \color{blue}{-0.12900613773279798}\right) \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.954929658551372, x, {x}^{3} \cdot -0.12900613773279798\right)} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 49.6%

   \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
6. Step-by-step derivation

   1. *-commutative 49.6%

      \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]

7. Simplified 49.6%

   \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]
8. Step-by-step derivation

   1. *-commutative 49.6%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]

   2. add-sqr-sqrt 22.6%

      \[\leadsto \color{blue}{\sqrt{0.954929658551372 \cdot x} \cdot \sqrt{0.954929658551372 \cdot x}} \]

   3. sqrt-unprod 26.3%

      \[\leadsto \color{blue}{\sqrt{\left(0.954929658551372 \cdot x\right) \cdot \left(0.954929658551372 \cdot x\right)}} \]

   4. *-commutative 26.3%

      \[\leadsto \sqrt{\left(0.954929658551372 \cdot x\right) \cdot \color{blue}{\left(x \cdot 0.954929658551372\right)}} \]

   5. *-commutative 26.3%

      \[\leadsto \sqrt{\color{blue}{\left(x \cdot 0.954929658551372\right)} \cdot \left(x \cdot 0.954929658551372\right)} \]

   6. swap-sqr 26.3%

      \[\leadsto \sqrt{\color{blue}{\left(x \cdot x\right) \cdot \left(0.954929658551372 \cdot 0.954929658551372\right)}} \]

   7. pow2 26.3%

      \[\leadsto \sqrt{\color{blue}{{x}^{2}} \cdot \left(0.954929658551372 \cdot 0.954929658551372\right)} \]

   8. metadata-eval 26.3%

      \[\leadsto \sqrt{{x}^{2} \cdot \color{blue}{0.9118906527810399}} \]

9. Applied egg-rr 26.3%

   \[\leadsto \color{blue}{\sqrt{{x}^{2} \cdot 0.9118906527810399}} \]

10. Taylor expanded in x around -inf 5.1%

    \[\leadsto \color{blue}{-0.954929658551372 \cdot x} \]
11. Step-by-step derivation

    1. *-commutative 5.1%

       \[\leadsto \color{blue}{x \cdot -0.954929658551372} \]

12. Simplified 5.1%

    \[\leadsto \color{blue}{x \cdot -0.954929658551372} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))