Rust f64::atanh

Percentage Accurate: 50.4% → 99.9%

Time: 1.9s

Alternatives: 5

Speedup: 3.4×

• Could not converge on a ground truth

  1. x = 1.7533746046869366e+195

Specification

Math

\[\tanh^{-1} x \]

FPCore

(FPCore (x)
  :precision binary64
  (atanh x))

C

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

Python

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

Julia

function code(x)
	return atanh(x)
end

MATLAB

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

Wolfram

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

Initial Program: 50.4% accurate, 1.0× speedup

Math

\[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]

FPCore

(FPCore (x)
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))

C

double code(double x) {
	return 0.5 * log1p(((2.0 * x) / (1.0 - x)));
}

Java

public static double code(double x) {
	return 0.5 * Math.log1p(((2.0 * x) / (1.0 - x)));
}

Python

def code(x):
	return 0.5 * math.log1p(((2.0 * x) / (1.0 - x)))

Julia

function code(x)
	return Float64(0.5 * log1p(Float64(Float64(2.0 * x) / Float64(1.0 - x))))
end

Wolfram

code[x_] := N[(0.5 * N[Log[1 + N[(N[(2.0 * x), $MachinePrecision] / N[(1.0 - x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.023:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\frac{\left|x\right| + \left|x\right|}{1 - \left|x\right|}\right)\\ \mathbf{else}:\\ \;\;\;\;\log 1\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (*
 (copysign 1.0 x)
 (if (<= (fabs x) 0.023)
   (* 0.5 (log1p (/ (+ (fabs x) (fabs x)) (- 1.0 (fabs x)))))
   (log 1.0))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.023) {
		tmp = 0.5 * log1p(((fabs(x) + fabs(x)) / (1.0 - fabs(x))));
	} else {
		tmp = log(1.0);
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.023) {
		tmp = 0.5 * Math.log1p(((Math.abs(x) + Math.abs(x)) / (1.0 - Math.abs(x))));
	} else {
		tmp = Math.log(1.0);
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.023:
		tmp = 0.5 * math.log1p(((math.fabs(x) + math.fabs(x)) / (1.0 - math.fabs(x))))
	else:
		tmp = math.log(1.0)
	return math.copysign(1.0, x) * tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.023)
		tmp = Float64(0.5 * log1p(Float64(Float64(abs(x) + abs(x)) / Float64(1.0 - abs(x)))));
	else
		tmp = log(1.0);
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 0.023], N[(0.5 * N[Log[1 + N[(N[(N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[1.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.023

   1. Initial program 50.4%

      \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{2 \cdot x}}{1 - x}\right) \]

      2. count-2-rev N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{x + x}}{1 - x}\right) \]

      3. lower-+.f64 50.4%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{\color{blue}{x + x}}{1 - x}\right) \]

   3. Applied rewrites 50.4%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{\color{blue}{x + x}}{1 - x}\right) \]

   if 0.023 < x

   1. Initial program 50.4%

      \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)} \]

      2. lift-log1p.f64 N/A

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

      3. log-pow-rev N/A

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

      4. lower-log.f64 N/A

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

      5. unpow1/2 N/A

         \[\leadsto \log \color{blue}{\left(\sqrt{1 + \frac{2 \cdot x}{1 - x}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \log \color{blue}{\left(\sqrt{1 + \frac{2 \cdot x}{1 - x}}\right)} \]

      7. +-commutative N/A

         \[\leadsto \log \left(\sqrt{\color{blue}{\frac{2 \cdot x}{1 - x} + 1}}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \log \left(\sqrt{\color{blue}{\frac{2 \cdot x}{1 - x}} + 1}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \log \left(\sqrt{\frac{\color{blue}{2 \cdot x}}{1 - x} + 1}\right) \]

      10. associate-/l* N/A

          \[\leadsto \log \left(\sqrt{\color{blue}{2 \cdot \frac{x}{1 - x}} + 1}\right) \]

      11. *-commutative N/A

          \[\leadsto \log \left(\sqrt{\color{blue}{\frac{x}{1 - x} \cdot 2} + 1}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \log \left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{1 - x}, 2, 1\right)}}\right) \]

      13. lower-/.f64 4.1%

          \[\leadsto \log \left(\sqrt{\mathsf{fma}\left(\color{blue}{\frac{x}{1 - x}}, 2, 1\right)}\right) \]

   3. Applied rewrites 4.1%

      \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(\frac{x}{1 - x}, 2, 1\right)}\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \log \color{blue}{\left(1 + x\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 4.4%

         \[\leadsto \log \left(1 + \color{blue}{x}\right) \]

   6. Applied rewrites 4.4%

      \[\leadsto \log \color{blue}{\left(1 + x\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \log 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 52.3%

      \[\leadsto \log 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.6% accurate, 0.7× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.023:\\ \;\;\;\;\mathsf{log1p}\left(\mathsf{fma}\left(\left|x\right|, \left|x\right|, \left|x\right|\right) \cdot 2\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;\log 1\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (*
 (copysign 1.0 x)
 (if (<= (fabs x) 0.023)
   (* (log1p (* (fma (fabs x) (fabs x) (fabs x)) 2.0)) 0.5)
   (log 1.0))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.023) {
		tmp = log1p((fma(fabs(x), fabs(x), fabs(x)) * 2.0)) * 0.5;
	} else {
		tmp = log(1.0);
	}
	return copysign(1.0, x) * tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.023)
		tmp = Float64(log1p(Float64(fma(abs(x), abs(x), abs(x)) * 2.0)) * 0.5);
	else
		tmp = log(1.0);
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 0.023], N[(N[Log[1 + N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[Abs[x], $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * 0.5), $MachinePrecision], N[Log[1.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.023

   1. Initial program 50.4%

      \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \cdot \frac{1}{2}} \]

      3. lower-*.f64 50.4%

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \cdot 0.5} \]

      4. lift-log1p.f64 N/A

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

      5. lower-log.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 4.1%

          \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{\frac{x}{1 - x}}, 2, 1\right)\right) \cdot 0.5 \]

   3. Applied rewrites 4.1%

      \[\leadsto \color{blue}{\log \left(\mathsf{fma}\left(\frac{x}{1 - x}, 2, 1\right)\right) \cdot 0.5} \]

   4. Taylor expanded in x around 0

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot \left(1 + x\right)}, 2, 1\right)\right) \cdot 0.5 \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. lower-+.f64 5.0%

         \[\leadsto \log \left(\mathsf{fma}\left(x \cdot \left(1 + \color{blue}{x}\right), 2, 1\right)\right) \cdot 0.5 \]

   6. Applied rewrites 5.0%

      \[\leadsto \log \left(\mathsf{fma}\left(\color{blue}{x \cdot \left(1 + x\right)}, 2, 1\right)\right) \cdot 0.5 \]
   7. Step-by-step derivation

      1. lift-log.f64 N/A

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

      2. lift-fma.f64 N/A

         \[\leadsto \log \color{blue}{\left(\left(x \cdot \left(1 + x\right)\right) \cdot 2 + 1\right)} \cdot \frac{1}{2} \]

      3. +-commutative N/A

         \[\leadsto \log \color{blue}{\left(1 + \left(x \cdot \left(1 + x\right)\right) \cdot 2\right)} \cdot \frac{1}{2} \]

      4. lower-log1p.f64 N/A

         \[\leadsto \color{blue}{\mathsf{log1p}\left(\left(x \cdot \left(1 + x\right)\right) \cdot 2\right)} \cdot \frac{1}{2} \]

      5. lower-*.f64 51.1%

         \[\leadsto \mathsf{log1p}\left(\color{blue}{\left(x \cdot \left(1 + x\right)\right) \cdot 2}\right) \cdot 0.5 \]

      6. lift-*.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\left(x \cdot \color{blue}{\left(1 + x\right)}\right) \cdot 2\right) \cdot \frac{1}{2} \]

      7. lift-+.f64 N/A

         \[\leadsto \mathsf{log1p}\left(\left(x \cdot \left(1 + \color{blue}{x}\right)\right) \cdot 2\right) \cdot \frac{1}{2} \]

      8. distribute-rgt-in N/A

         \[\leadsto \mathsf{log1p}\left(\left(1 \cdot x + \color{blue}{x \cdot x}\right) \cdot 2\right) \cdot \frac{1}{2} \]

      9. *-lft-identity N/A

         \[\leadsto \mathsf{log1p}\left(\left(x + \color{blue}{x} \cdot x\right) \cdot 2\right) \cdot \frac{1}{2} \]

      10. +-commutative N/A

          \[\leadsto \mathsf{log1p}\left(\left(x \cdot x + \color{blue}{x}\right) \cdot 2\right) \cdot \frac{1}{2} \]

      11. lower-fma.f64 51.1%

          \[\leadsto \mathsf{log1p}\left(\mathsf{fma}\left(x, \color{blue}{x}, x\right) \cdot 2\right) \cdot 0.5 \]

   8. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{fma}\left(x, x, x\right) \cdot 2\right)} \cdot 0.5 \]

   if 0.023 < x

   1. Initial program 50.4%

      \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)} \]

      2. lift-log1p.f64 N/A

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

      3. log-pow-rev N/A

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

      4. lower-log.f64 N/A

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

      5. unpow1/2 N/A

         \[\leadsto \log \color{blue}{\left(\sqrt{1 + \frac{2 \cdot x}{1 - x}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \log \color{blue}{\left(\sqrt{1 + \frac{2 \cdot x}{1 - x}}\right)} \]

      7. +-commutative N/A

         \[\leadsto \log \left(\sqrt{\color{blue}{\frac{2 \cdot x}{1 - x} + 1}}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \log \left(\sqrt{\color{blue}{\frac{2 \cdot x}{1 - x}} + 1}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \log \left(\sqrt{\frac{\color{blue}{2 \cdot x}}{1 - x} + 1}\right) \]

      10. associate-/l* N/A

          \[\leadsto \log \left(\sqrt{\color{blue}{2 \cdot \frac{x}{1 - x}} + 1}\right) \]

      11. *-commutative N/A

          \[\leadsto \log \left(\sqrt{\color{blue}{\frac{x}{1 - x} \cdot 2} + 1}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \log \left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{1 - x}, 2, 1\right)}}\right) \]

      13. lower-/.f64 4.1%

          \[\leadsto \log \left(\sqrt{\mathsf{fma}\left(\color{blue}{\frac{x}{1 - x}}, 2, 1\right)}\right) \]

   3. Applied rewrites 4.1%

      \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(\frac{x}{1 - x}, 2, 1\right)}\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \log \color{blue}{\left(1 + x\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 4.4%

         \[\leadsto \log \left(1 + \color{blue}{x}\right) \]

   6. Applied rewrites 4.4%

      \[\leadsto \log \color{blue}{\left(1 + x\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \log 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 52.3%

      \[\leadsto \log 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.1% accurate, 0.8× speedup

Math

\[\mathsf{copysign}\left(1, x\right) \cdot \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.023:\\ \;\;\;\;0.5 \cdot \mathsf{log1p}\left(\left|x\right| \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\log 1\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (*
 (copysign 1.0 x)
 (if (<= (fabs x) 0.023) (* 0.5 (log1p (* (fabs x) 2.0))) (log 1.0))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.023) {
		tmp = 0.5 * log1p((fabs(x) * 2.0));
	} else {
		tmp = log(1.0);
	}
	return copysign(1.0, x) * tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.023) {
		tmp = 0.5 * Math.log1p((Math.abs(x) * 2.0));
	} else {
		tmp = Math.log(1.0);
	}
	return Math.copySign(1.0, x) * tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.023:
		tmp = 0.5 * math.log1p((math.fabs(x) * 2.0))
	else:
		tmp = math.log(1.0)
	return math.copysign(1.0, x) * tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.023)
		tmp = Float64(0.5 * log1p(Float64(abs(x) * 2.0)));
	else
		tmp = log(1.0);
	end
	return Float64(copysign(1.0, x) * tmp)
end

Wolfram

code[x_] := N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[Abs[x], $MachinePrecision], 0.023], N[(0.5 * N[Log[1 + N[(N[Abs[x], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[Log[1.0], $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 0.023

   1. Initial program 50.4%

      \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{2 \cdot x}}{1 - x}\right) \]

      2. count-2-rev N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(\frac{\color{blue}{x + x}}{1 - x}\right) \]

      3. lower-+.f64 50.4%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{\color{blue}{x + x}}{1 - x}\right) \]

   3. Applied rewrites 50.4%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\frac{\color{blue}{x + x}}{1 - x}\right) \]

   4. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{x \cdot \left(2 + 2 \cdot x\right)}\right) \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(x \cdot \color{blue}{\left(2 + 2 \cdot x\right)}\right) \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \mathsf{log1p}\left(x \cdot \left(2 + \color{blue}{2 \cdot x}\right)\right) \]

      3. lower-*.f64 51.1%

         \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot \left(2 + 2 \cdot \color{blue}{x}\right)\right) \]

   6. Applied rewrites 51.1%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(\color{blue}{x \cdot \left(2 + 2 \cdot x\right)}\right) \]

   7. Taylor expanded in x around 0

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot 2\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 50.2%

      \[\leadsto 0.5 \cdot \mathsf{log1p}\left(x \cdot 2\right) \]

   if 0.023 < x

   1. Initial program 50.4%

      \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)} \]

      2. lift-log1p.f64 N/A

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

      3. log-pow-rev N/A

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

      4. lower-log.f64 N/A

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

      5. unpow1/2 N/A

         \[\leadsto \log \color{blue}{\left(\sqrt{1 + \frac{2 \cdot x}{1 - x}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \log \color{blue}{\left(\sqrt{1 + \frac{2 \cdot x}{1 - x}}\right)} \]

      7. +-commutative N/A

         \[\leadsto \log \left(\sqrt{\color{blue}{\frac{2 \cdot x}{1 - x} + 1}}\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \log \left(\sqrt{\color{blue}{\frac{2 \cdot x}{1 - x}} + 1}\right) \]

      9. lift-*.f64 N/A

         \[\leadsto \log \left(\sqrt{\frac{\color{blue}{2 \cdot x}}{1 - x} + 1}\right) \]

      10. associate-/l* N/A

          \[\leadsto \log \left(\sqrt{\color{blue}{2 \cdot \frac{x}{1 - x}} + 1}\right) \]

      11. *-commutative N/A

          \[\leadsto \log \left(\sqrt{\color{blue}{\frac{x}{1 - x} \cdot 2} + 1}\right) \]

      12. lower-fma.f64 N/A

          \[\leadsto \log \left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{1 - x}, 2, 1\right)}}\right) \]

      13. lower-/.f64 4.1%

          \[\leadsto \log \left(\sqrt{\mathsf{fma}\left(\color{blue}{\frac{x}{1 - x}}, 2, 1\right)}\right) \]

   3. Applied rewrites 4.1%

      \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(\frac{x}{1 - x}, 2, 1\right)}\right)} \]

   4. Taylor expanded in x around 0

      \[\leadsto \log \color{blue}{\left(1 + x\right)} \]
   5. Step-by-step derivation

      1. lower-+.f64 4.4%

         \[\leadsto \log \left(1 + \color{blue}{x}\right) \]

   6. Applied rewrites 4.4%

      \[\leadsto \log \color{blue}{\left(1 + x\right)} \]

   7. Taylor expanded in x around 0

      \[\leadsto \log 1 \]
   8. Step-by-step derivation

   9. Applied rewrites 52.3%

      \[\leadsto \log 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 52.3% accurate, 3.4× speedup

Math

\[\log 1 \]

FPCore

(FPCore (x)
  :precision binary64
  (log 1.0))

C

double code(double x) {
	return log(1.0);
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = log(1.0d0)
end function

Java

public static double code(double x) {
	return Math.log(1.0);
}

Python

def code(x):
	return math.log(1.0)

Julia

function code(x)
	return log(1.0)
end

MATLAB

function tmp = code(x)
	tmp = log(1.0);
end

Wolfram

code[x_] := N[Log[1.0], $MachinePrecision]

Derivation

1. Initial program 50.4%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right)} \]

   2. lift-log1p.f64 N/A

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

   3. log-pow-rev N/A

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

   4. lower-log.f64 N/A

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

   5. unpow1/2 N/A

      \[\leadsto \log \color{blue}{\left(\sqrt{1 + \frac{2 \cdot x}{1 - x}}\right)} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \log \color{blue}{\left(\sqrt{1 + \frac{2 \cdot x}{1 - x}}\right)} \]

   7. +-commutative N/A

      \[\leadsto \log \left(\sqrt{\color{blue}{\frac{2 \cdot x}{1 - x} + 1}}\right) \]

   8. lift-/.f64 N/A

      \[\leadsto \log \left(\sqrt{\color{blue}{\frac{2 \cdot x}{1 - x}} + 1}\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \log \left(\sqrt{\frac{\color{blue}{2 \cdot x}}{1 - x} + 1}\right) \]

   10. associate-/l* N/A

       \[\leadsto \log \left(\sqrt{\color{blue}{2 \cdot \frac{x}{1 - x}} + 1}\right) \]

   11. *-commutative N/A

       \[\leadsto \log \left(\sqrt{\color{blue}{\frac{x}{1 - x} \cdot 2} + 1}\right) \]

   12. lower-fma.f64 N/A

       \[\leadsto \log \left(\sqrt{\color{blue}{\mathsf{fma}\left(\frac{x}{1 - x}, 2, 1\right)}}\right) \]

   13. lower-/.f64 4.1%

       \[\leadsto \log \left(\sqrt{\mathsf{fma}\left(\color{blue}{\frac{x}{1 - x}}, 2, 1\right)}\right) \]

3. Applied rewrites 4.1%

   \[\leadsto \color{blue}{\log \left(\sqrt{\mathsf{fma}\left(\frac{x}{1 - x}, 2, 1\right)}\right)} \]

4. Taylor expanded in x around 0

   \[\leadsto \log \color{blue}{\left(1 + x\right)} \]
5. Step-by-step derivation

   1. lower-+.f64 4.4%

      \[\leadsto \log \left(1 + \color{blue}{x}\right) \]

6. Applied rewrites 4.4%

   \[\leadsto \log \color{blue}{\left(1 + x\right)} \]

7. Taylor expanded in x around 0

   \[\leadsto \log 1 \]
8. Step-by-step derivation

9. Applied rewrites 52.3%

   \[\leadsto \log 1 \]
10. Add Preprocessing

Alternative 5: 4.6% accurate, 5.0× speedup

Math

\[\frac{1}{x} \]

FPCore

(FPCore (x)
  :precision binary64
  (/ 1.0 x))

C

double code(double x) {
	return 1.0 / x;
}

Fortran

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = 1.0d0 / x
end function

Java

public static double code(double x) {
	return 1.0 / x;
}

Python

def code(x):
	return 1.0 / x

Julia

function code(x)
	return Float64(1.0 / x)
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / x;
end

Wolfram

code[x_] := N[(1.0 / x), $MachinePrecision]

Derivation

1. Initial program 50.4%

   \[0.5 \cdot \mathsf{log1p}\left(\frac{2 \cdot x}{1 - x}\right) \]
2. Step-by-step derivation

   1. lift-log1p.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. add-to-fraction N/A

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

   4. log-div N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\log \left(\left|1 \cdot \left(1 - x\right) + 2 \cdot x\right|\right) - \log \left(\left|1 - x\right|\right)\right)} \]

   5. diff-log N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\log \left(\frac{\left|1 \cdot \left(1 - x\right) + 2 \cdot x\right|}{\left|1 - x\right|}\right)} \]

   6. fabs-div N/A

      \[\leadsto \frac{1}{2} \cdot \log \color{blue}{\left(\left|\frac{1 \cdot \left(1 - x\right) + 2 \cdot x}{1 - x}\right|\right)} \]

   7. add-to-fraction N/A

      \[\leadsto \frac{1}{2} \cdot \log \left(\left|\color{blue}{1 + \frac{2 \cdot x}{1 - x}}\right|\right) \]

   8. lift-/.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \log \left(\left|1 + \color{blue}{\frac{2 \cdot x}{1 - x}}\right|\right) \]

   9. lower-log.f64 N/A

      \[\leadsto \frac{1}{2} \cdot \color{blue}{\log \left(\left|1 + \frac{2 \cdot x}{1 - x}\right|\right)} \]

   10. add-flip N/A

       \[\leadsto \frac{1}{2} \cdot \log \left(\left|\color{blue}{1 - \left(\mathsf{neg}\left(\frac{2 \cdot x}{1 - x}\right)\right)}\right|\right) \]

   11. fabs-sub N/A

       \[\leadsto \frac{1}{2} \cdot \log \color{blue}{\left(\left|\left(\mathsf{neg}\left(\frac{2 \cdot x}{1 - x}\right)\right) - 1\right|\right)} \]

   12. lower-fabs.f64 N/A

       \[\leadsto \frac{1}{2} \cdot \log \color{blue}{\left(\left|\left(\mathsf{neg}\left(\frac{2 \cdot x}{1 - x}\right)\right) - 1\right|\right)} \]

   13. sub-flip N/A

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

   14. metadata-eval N/A

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

3. Applied rewrites 51.7%

   \[\leadsto 0.5 \cdot \color{blue}{\log \left(\left|\mathsf{fma}\left(-2, \frac{x}{1 - x}, -1\right)\right|\right)} \]

4. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{x}} \]
5. Step-by-step derivation

   1. lower-/.f64 4.6%

      \[\leadsto \frac{1}{\color{blue}{x}} \]

6. Applied rewrites 4.6%

   \[\leadsto \color{blue}{\frac{1}{x}} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2025313 -o setup:search
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))