Rust f64::asinh

Percentage Accurate: 29.8% → 99.8%

Time: 6.2s

Alternatives: 4

Speedup: 1.9×

Specification

Math

\[\begin{array}{l} \\ \sinh^{-1} x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (asinh x))

C

double code(double x) {
	return asinh(x);
}

Python

def code(x):
	return math.asinh(x)

Julia

function code(x)
	return asinh(x)
end

MATLAB

function tmp = code(x)
	tmp = asinh(x);
end

Wolfram

code[x_] := N[ArcSinh[x], $MachinePrecision]

Initial Program: 29.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))

C

double code(double x) {
	return copysign(log((fabs(x) + sqrt(((x * x) + 1.0)))), x);
}

Java

public static double code(double x) {
	return Math.copySign(Math.log((Math.abs(x) + Math.sqrt(((x * x) + 1.0)))), x);
}

Python

def code(x):
	return math.copysign(math.log((math.fabs(x) + math.sqrt(((x * x) + 1.0)))), x)

Julia

function code(x)
	return copysign(log(Float64(abs(x) + sqrt(Float64(Float64(x * x) + 1.0)))), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(log((abs(x) + sqrt(((x * x) + 1.0)))));
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[Log[N[(N[Abs[x], $MachinePrecision] + N[Sqrt[N[(N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Alternative 1: 99.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\sinh^{-1} x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (copysign (asinh x) x))

C

double code(double x) {
	return copysign(asinh(x), x);
}

Python

def code(x):
	return math.copysign(math.asinh(x), x)

Julia

function code(x)
	return copysign(asinh(x), x)
end

MATLAB

function tmp = code(x)
	tmp = sign(x) * abs(asinh(x));
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[ArcSinh[x], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Derivation

1. Initial program 29.5%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}, x\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}, x\right) \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\sqrt{x \cdot x + 1}}\right), x\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{x \cdot x + 1}}\right), x\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{x \cdot x} + 1}\right), x\right) \]

   6. sqr-abs-rev N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{\left|x\right| \cdot \left|x\right|} + 1}\right), x\right) \]

   7. lift-fabs.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{\left|x\right|} \cdot \left|x\right| + 1}\right), x\right) \]

   8. lift-fabs.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\left|x\right| \cdot \color{blue}{\left|x\right|} + 1}\right), x\right) \]

   9. asinh-def-rev N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(\left|x\right|\right)}, x\right) \]

   10. lower-asinh.f64 99.8

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(\left|x\right|\right)}, x\right) \]

   11. lift-fabs.f64 N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left(\left|x\right|\right)}, x\right) \]

   12. rem-sqrt-square-rev N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left(\sqrt{x \cdot x}\right)}, x\right) \]

   13. pow2 N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\sqrt{\color{blue}{{x}^{2}}}\right), x\right) \]

   14. sqrt-pow1 N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left({x}^{\left(\frac{2}{2}\right)}\right)}, x\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left({x}^{\color{blue}{1}}\right), x\right) \]

   16. unpow1 99.8

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{x}, x\right) \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} x, x\right)} \]
5. Add Preprocessing

Alternative 2: 51.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.075 \cdot x, x, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144\right) - 0.16666666666666666\right) \cdot \left(x \cdot x\right), x, x\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (copysign
  (fma
   (*
    (fma
     (* 0.075 x)
     x
     (- (* (* x x) (* (* x x) -0.044642857142857144)) 0.16666666666666666))
    (* x x))
   x
   x)
  x))

C

double code(double x) {
	return copysign(fma((fma((0.075 * x), x, (((x * x) * ((x * x) * -0.044642857142857144)) - 0.16666666666666666)) * (x * x)), x, x), x);
}

Julia

function code(x)
	return copysign(fma(Float64(fma(Float64(0.075 * x), x, Float64(Float64(Float64(x * x) * Float64(Float64(x * x) * -0.044642857142857144)) - 0.16666666666666666)) * Float64(x * x)), x, x), x)
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[(N[(N[(N[(0.075 * x), $MachinePrecision] * x + N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.044642857142857144), $MachinePrecision]), $MachinePrecision] - 0.16666666666666666), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Derivation

1. Initial program 29.5%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}, x\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}, x\right) \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\sqrt{x \cdot x + 1}}\right), x\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{x \cdot x + 1}}\right), x\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{x \cdot x} + 1}\right), x\right) \]

   6. sqr-abs-rev N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{\left|x\right| \cdot \left|x\right|} + 1}\right), x\right) \]

   7. lift-fabs.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{\left|x\right|} \cdot \left|x\right| + 1}\right), x\right) \]

   8. lift-fabs.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\left|x\right| \cdot \color{blue}{\left|x\right|} + 1}\right), x\right) \]

   9. asinh-def-rev N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(\left|x\right|\right)}, x\right) \]

   10. lower-asinh.f64 99.8

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(\left|x\right|\right)}, x\right) \]

   11. lift-fabs.f64 N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left(\left|x\right|\right)}, x\right) \]

   12. rem-sqrt-square-rev N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left(\sqrt{x \cdot x}\right)}, x\right) \]

   13. pow2 N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\sqrt{\color{blue}{{x}^{2}}}\right), x\right) \]

   14. sqrt-pow1 N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left({x}^{\left(\frac{2}{2}\right)}\right)}, x\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left({x}^{\color{blue}{1}}\right), x\right) \]

   16. unpow1 99.8

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{x}, x\right) \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} x, x\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right)}, x\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{copysign}\left(x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + 1\right)}, x\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)\right) + x \cdot 1}, x\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right)} + x \cdot 1, x\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{copysign}\left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + x \cdot 1, x\right) \]

   5. cube-mult N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{3}} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + x \cdot 1, x\right) \]

   6. *-rgt-identity N/A

      \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot \left({x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}\right) + \color{blue}{x}, x\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{3}, {x}^{2} \cdot \left(\frac{3}{40} + \frac{-5}{112} \cdot {x}^{2}\right) - \frac{1}{6}, x\right)}, x\right) \]

7. Applied rewrites 52.2%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.075 \cdot x, x, {x}^{4} \cdot -0.044642857142857144 - 0.16666666666666666\right), x\right)}, x\right) \]
8. Step-by-step derivation

9. Applied rewrites 52.2%

   \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.075 \cdot x, x, -0.044642857142857144 \cdot {x}^{4} - 0.16666666666666666\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right), x\right) \]
10. Step-by-step derivation

11. Applied rewrites 52.2%

    \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.075 \cdot x, x, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot -0.044642857142857144\right) - 0.16666666666666666\right) \cdot \left(x \cdot x\right), x, x\right), x\right) \]
12. Add Preprocessing

Alternative 3: 51.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.075 - 0.16666666666666666\right) \cdot \left(x \cdot x\right), x, x\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (copysign (fma (* (- (* (* x x) 0.075) 0.16666666666666666) (* x x)) x x) x))

C

double code(double x) {
	return copysign(fma(((((x * x) * 0.075) - 0.16666666666666666) * (x * x)), x, x), x);
}

Julia

function code(x)
	return copysign(fma(Float64(Float64(Float64(Float64(x * x) * 0.075) - 0.16666666666666666) * Float64(x * x)), x, x), x)
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.075), $MachinePrecision] - 0.16666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Derivation

1. Initial program 29.5%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}, x\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}, x\right) \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\sqrt{x \cdot x + 1}}\right), x\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{x \cdot x + 1}}\right), x\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{x \cdot x} + 1}\right), x\right) \]

   6. sqr-abs-rev N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{\left|x\right| \cdot \left|x\right|} + 1}\right), x\right) \]

   7. lift-fabs.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{\left|x\right|} \cdot \left|x\right| + 1}\right), x\right) \]

   8. lift-fabs.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\left|x\right| \cdot \color{blue}{\left|x\right|} + 1}\right), x\right) \]

   9. asinh-def-rev N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(\left|x\right|\right)}, x\right) \]

   10. lower-asinh.f64 99.8

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(\left|x\right|\right)}, x\right) \]

   11. lift-fabs.f64 N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left(\left|x\right|\right)}, x\right) \]

   12. rem-sqrt-square-rev N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left(\sqrt{x \cdot x}\right)}, x\right) \]

   13. pow2 N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\sqrt{\color{blue}{{x}^{2}}}\right), x\right) \]

   14. sqrt-pow1 N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left({x}^{\left(\frac{2}{2}\right)}\right)}, x\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left({x}^{\color{blue}{1}}\right), x\right) \]

   16. unpow1 99.8

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{x}, x\right) \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} x, x\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)\right)}, x\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{copysign}\left(x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) + 1\right)}, x\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)\right) + x \cdot 1}, x\right) \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right)} + x \cdot 1, x\right) \]

   4. unpow2 N/A

      \[\leadsto \mathsf{copysign}\left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) + x \cdot 1, x\right) \]

   5. cube-mult N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{3}} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) + x \cdot 1, x\right) \]

   6. *-rgt-identity N/A

      \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot \left(\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}\right) + \color{blue}{x}, x\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{3}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right)}, x\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\color{blue}{{x}^{3}}, \frac{3}{40} \cdot {x}^{2} - \frac{1}{6}, x\right), x\right) \]

   9. lower--.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left({x}^{3}, \color{blue}{\frac{3}{40} \cdot {x}^{2} - \frac{1}{6}}, x\right), x\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left({x}^{3}, \color{blue}{\frac{3}{40} \cdot {x}^{2}} - \frac{1}{6}, x\right), x\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left({x}^{3}, \frac{3}{40} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{6}, x\right), x\right) \]

   12. lower-*.f64 52.2

       \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left({x}^{3}, 0.075 \cdot \color{blue}{\left(x \cdot x\right)} - 0.16666666666666666, x\right), x\right) \]

7. Applied rewrites 52.2%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{3}, 0.075 \cdot \left(x \cdot x\right) - 0.16666666666666666, x\right)}, x\right) \]
8. Step-by-step derivation

9. Applied rewrites 52.2%

   \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.075 - 0.16666666666666666\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right), x\right) \]
10. Add Preprocessing

Alternative 4: 51.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{copysign}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (copysign (* (fma (* -0.16666666666666666 x) x 1.0) x) x))

C

double code(double x) {
	return copysign((fma((-0.16666666666666666 * x), x, 1.0) * x), x);
}

Julia

function code(x)
	return copysign(Float64(fma(Float64(-0.16666666666666666 * x), x, 1.0) * x), x)
end

Wolfram

code[x_] := N[With[{TMP1 = Abs[N[(N[(N[(-0.16666666666666666 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]

Derivation

1. Initial program 29.5%

   \[\mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right), x\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-log.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\log \left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}, x\right) \]

   2. lift-+.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \color{blue}{\left(\left|x\right| + \sqrt{x \cdot x + 1}\right)}, x\right) \]

   3. lift-sqrt.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \color{blue}{\sqrt{x \cdot x + 1}}\right), x\right) \]

   4. lift-+.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{x \cdot x + 1}}\right), x\right) \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{x \cdot x} + 1}\right), x\right) \]

   6. sqr-abs-rev N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{\left|x\right| \cdot \left|x\right|} + 1}\right), x\right) \]

   7. lift-fabs.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\color{blue}{\left|x\right|} \cdot \left|x\right| + 1}\right), x\right) \]

   8. lift-fabs.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\log \left(\left|x\right| + \sqrt{\left|x\right| \cdot \color{blue}{\left|x\right|} + 1}\right), x\right) \]

   9. asinh-def-rev N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(\left|x\right|\right)}, x\right) \]

   10. lower-asinh.f64 99.8

       \[\leadsto \mathsf{copysign}\left(\color{blue}{\sinh^{-1} \left(\left|x\right|\right)}, x\right) \]

   11. lift-fabs.f64 N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left(\left|x\right|\right)}, x\right) \]

   12. rem-sqrt-square-rev N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left(\sqrt{x \cdot x}\right)}, x\right) \]

   13. pow2 N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left(\sqrt{\color{blue}{{x}^{2}}}\right), x\right) \]

   14. sqrt-pow1 N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{\left({x}^{\left(\frac{2}{2}\right)}\right)}, x\right) \]

   15. metadata-eval N/A

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \left({x}^{\color{blue}{1}}\right), x\right) \]

   16. unpow1 99.8

       \[\leadsto \mathsf{copysign}\left(\sinh^{-1} \color{blue}{x}, x\right) \]

4. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\mathsf{copysign}\left(\sinh^{-1} x, x\right)} \]

5. Taylor expanded in x around 0

   \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(1 + \frac{-1}{6} \cdot {x}^{2}\right)}, x\right) \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \mathsf{copysign}\left(x \cdot \color{blue}{\left(\frac{-1}{6} \cdot {x}^{2} + 1\right)}, x\right) \]

   2. distribute-lft-in N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{x \cdot \left(\frac{-1}{6} \cdot {x}^{2}\right) + x \cdot 1}, x\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{copysign}\left(x \cdot \color{blue}{\left({x}^{2} \cdot \frac{-1}{6}\right)} + x \cdot 1, x\right) \]

   4. associate-*r* N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\left(x \cdot {x}^{2}\right) \cdot \frac{-1}{6}} + x \cdot 1, x\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{copysign}\left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \frac{-1}{6} + x \cdot 1, x\right) \]

   6. cube-mult N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{{x}^{3}} \cdot \frac{-1}{6} + x \cdot 1, x\right) \]

   7. *-rgt-identity N/A

      \[\leadsto \mathsf{copysign}\left({x}^{3} \cdot \frac{-1}{6} + \color{blue}{x}, x\right) \]

   8. lower-fma.f64 N/A

      \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{3}, \frac{-1}{6}, x\right)}, x\right) \]

   9. lower-pow.f64 52.0

      \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(\color{blue}{{x}^{3}}, -0.16666666666666666, x\right), x\right) \]

7. Applied rewrites 52.0%

   \[\leadsto \mathsf{copysign}\left(\color{blue}{\mathsf{fma}\left({x}^{3}, -0.16666666666666666, x\right)}, x\right) \]
8. Step-by-step derivation

9. Applied rewrites 52.0%

   \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot \left(x \cdot x\right), \color{blue}{x}, x\right), x\right) \]
10. Step-by-step derivation

11. Applied rewrites 52.0%

    \[\leadsto \mathsf{copysign}\left(\mathsf{fma}\left(-0.16666666666666666 \cdot x, x, 1\right) \cdot \color{blue}{x}, x\right) \]
12. Add Preprocessing

Developer Target 1: 100.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ \mathsf{copysign}\left(\mathsf{log1p}\left(\left|x\right| + \frac{\left|x\right|}{\mathsf{hypot}\left(1, t\_0\right) + t\_0}\right), x\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x))))
   (copysign (log1p (+ (fabs x) (/ (fabs x) (+ (hypot 1.0 t_0) t_0)))) x)))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	return copysign(log1p((fabs(x) + (fabs(x) / (hypot(1.0, t_0) + t_0)))), x);
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	return Math.copySign(Math.log1p((Math.abs(x) + (Math.abs(x) / (Math.hypot(1.0, t_0) + t_0)))), x);
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	return math.copysign(math.log1p((math.fabs(x) + (math.fabs(x) / (math.hypot(1.0, t_0) + t_0)))), x)

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	return copysign(log1p(Float64(abs(x) + Float64(abs(x) / Float64(hypot(1.0, t_0) + t_0)))), x)
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[With[{TMP1 = Abs[N[Log[1 + N[(N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] / N[(N[Sqrt[1.0 ^ 2 + t$95$0 ^ 2], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]]

Reproduce

herbie shell --seed 2024346 
(FPCore (x)
  :name "Rust f64::asinh"
  :precision binary64

  :alt
  (! :herbie-platform default (let* ((ax (fabs x)) (ix (/ 1 ax))) (copysign (log1p (+ ax (/ ax (+ (hypot 1 ix) ix)))) x)))

  (copysign (log (+ (fabs x) (sqrt (+ (* x x) 1.0)))) x))