Rust f64::atanh

Percentage Accurate: 100.0% → 99.7%

Time: 5.2s

Alternatives: 7

Speedup: N/A×

Specification

Math

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

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: 100.0% accurate, 1.0× speedup

Math

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

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: 99.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), 0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (fma x (* x (fma (* x x) 0.14285714285714285 0.2)) 0.3333333333333333)
  (* x (* x x))
  x))

C

double code(double x) {
	return fma(fma(x, (x * fma((x * x), 0.14285714285714285, 0.2)), 0.3333333333333333), (x * (x * x)), x);
}

Julia

function code(x)
	return fma(fma(x, Float64(x * fma(Float64(x * x), 0.14285714285714285, 0.2)), 0.3333333333333333), Float64(x * Float64(x * x)), x)
end

Wolfram

code[x_] := N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.14285714285714285 + 0.2), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right) + 1\right)} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right), 1\right)} \]

   4. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right), 1\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right), 1\right) \]

   6. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) + \frac{1}{3}}, 1\right) \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{5} + \frac{1}{7} \cdot {x}^{2}, \frac{1}{3}\right)}, 1\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5} + \frac{1}{7} \cdot {x}^{2}, \frac{1}{3}\right), 1\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5} + \frac{1}{7} \cdot {x}^{2}, \frac{1}{3}\right), 1\right) \]

   10. +-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{7} \cdot {x}^{2} + \frac{1}{5}}, \frac{1}{3}\right), 1\right) \]

   11. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{7}} + \frac{1}{5}, \frac{1}{3}\right), 1\right) \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{7}, \frac{1}{5}\right)}, \frac{1}{3}\right), 1\right) \]

   13. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{7}, \frac{1}{5}\right), \frac{1}{3}\right), 1\right) \]

   14. *-lowering-*.f64 100.0

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.14285714285714285, 0.2\right), 0.3333333333333333\right), 1\right) \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), 0.3333333333333333\right), 1\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{7} + \frac{1}{5}\right) + \frac{1}{3}\right)\right) \cdot x + 1 \cdot x} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{7} + \frac{1}{5}\right) + \frac{1}{3}\right) \cdot \left(x \cdot x\right)\right)} \cdot x + 1 \cdot x \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{7} + \frac{1}{5}\right) + \frac{1}{3}\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + 1 \cdot x \]

   4. pow3 N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{7} + \frac{1}{5}\right) + \frac{1}{3}\right) \cdot \color{blue}{{x}^{3}} + 1 \cdot x \]

   5. *-lft-identity N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{7} + \frac{1}{5}\right) + \frac{1}{3}\right) \cdot {x}^{3} + \color{blue}{x} \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{7} + \frac{1}{5}\right) + \frac{1}{3}, {x}^{3}, x\right)} \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{7} + \frac{1}{5}\right)\right)} + \frac{1}{3}, {x}^{3}, x\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{7} + \frac{1}{5}\right), \frac{1}{3}\right)}, {x}^{3}, x\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{7} + \frac{1}{5}\right)}, \frac{1}{3}\right), {x}^{3}, x\right) \]

   10. accelerator-lowering-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \frac{1}{7}, \frac{1}{5}\right)}, \frac{1}{3}\right), {x}^{3}, x\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{7}, \frac{1}{5}\right), \frac{1}{3}\right), {x}^{3}, x\right) \]

   12. cube-unmult N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{7}, \frac{1}{5}\right), \frac{1}{3}\right), \color{blue}{x \cdot \left(x \cdot x\right)}, x\right) \]

   13. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{7}, \frac{1}{5}\right), \frac{1}{3}\right), \color{blue}{x \cdot \left(x \cdot x\right)}, x\right) \]

   14. *-lowering-*.f64 100.0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), 0.3333333333333333\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), 0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]
8. Add Preprocessing

Alternative 2: 99.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), 0.3333333333333333\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (fma
   (* x x)
   (fma (* x x) (fma (* x x) 0.14285714285714285 0.2) 0.3333333333333333)
   1.0)))

C

double code(double x) {
	return x * fma((x * x), fma((x * x), fma((x * x), 0.14285714285714285, 0.2), 0.3333333333333333), 1.0);
}

Julia

function code(x)
	return Float64(x * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), 0.14285714285714285, 0.2), 0.3333333333333333), 1.0))
end

Wolfram

code[x_] := N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.14285714285714285 + 0.2), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right)\right) + 1\right)} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right), 1\right)} \]

   4. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right), 1\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right), 1\right) \]

   6. +-commutative N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{7} \cdot {x}^{2}\right) + \frac{1}{3}}, 1\right) \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{5} + \frac{1}{7} \cdot {x}^{2}, \frac{1}{3}\right)}, 1\right) \]

   8. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5} + \frac{1}{7} \cdot {x}^{2}, \frac{1}{3}\right), 1\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5} + \frac{1}{7} \cdot {x}^{2}, \frac{1}{3}\right), 1\right) \]

   10. +-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{7} \cdot {x}^{2} + \frac{1}{5}}, \frac{1}{3}\right), 1\right) \]

   11. *-commutative N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{7}} + \frac{1}{5}, \frac{1}{3}\right), 1\right) \]

   12. accelerator-lowering-fma.f64 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{7}, \frac{1}{5}\right)}, \frac{1}{3}\right), 1\right) \]

   13. unpow2 N/A

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{7}, \frac{1}{5}\right), \frac{1}{3}\right), 1\right) \]

   14. *-lowering-*.f64 100.0

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.14285714285714285, 0.2\right), 0.3333333333333333\right), 1\right) \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.14285714285714285, 0.2\right), 0.3333333333333333\right), 1\right)} \]
6. Add Preprocessing

Alternative 3: 99.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.2, 0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (fma x (* x 0.2) 0.3333333333333333) (* x (* x x)) x))

C

double code(double x) {
	return fma(fma(x, (x * 0.2), 0.3333333333333333), (x * (x * x)), x);
}

Julia

function code(x)
	return fma(fma(x, Float64(x * 0.2), 0.3333333333333333), Float64(x * Float64(x * x)), x)
end

Wolfram

code[x_] := N[(N[(x * N[(x * 0.2), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) + 1\right)} \]

   3. accelerator-lowering-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{5}, \frac{1}{3}\right)}, 1\right) \]

   9. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5}, \frac{1}{3}\right), 1\right) \]

   10. *-lowering-*.f64 100.0

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.2, 0.3333333333333333\right), 1\right) \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right), 1\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{1}{3}\right)\right) \cdot x + 1 \cdot x} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{1}{3}\right) \cdot \left(x \cdot x\right)\right)} \cdot x + 1 \cdot x \]

   3. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{1}{3}\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} + 1 \cdot x \]

   4. pow3 N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{1}{3}\right) \cdot \color{blue}{{x}^{3}} + 1 \cdot x \]

   5. *-lft-identity N/A

      \[\leadsto \left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{1}{3}\right) \cdot {x}^{3} + \color{blue}{x} \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{5} + \frac{1}{3}, {x}^{3}, x\right)} \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(x \cdot \frac{1}{5}\right)} + \frac{1}{3}, {x}^{3}, x\right) \]

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{5}, \frac{1}{3}\right)}, {x}^{3}, x\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{5}}, \frac{1}{3}\right), {x}^{3}, x\right) \]

   10. cube-unmult N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{1}{5}, \frac{1}{3}\right), \color{blue}{x \cdot \left(x \cdot x\right)}, x\right) \]

   11. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{1}{5}, \frac{1}{3}\right), \color{blue}{x \cdot \left(x \cdot x\right)}, x\right) \]

   12. *-lowering-*.f64 100.0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.2, 0.3333333333333333\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.2, 0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]
8. Add Preprocessing

Alternative 4: 99.6% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (fma (* x x) (fma (* x x) 0.2 0.3333333333333333) 1.0)))

C

double code(double x) {
	return x * fma((x * x), fma((x * x), 0.2, 0.3333333333333333), 1.0);
}

Julia

function code(x)
	return Float64(x * fma(Float64(x * x), fma(Float64(x * x), 0.2, 0.3333333333333333), 1.0))
end

Wolfram

code[x_] := N[(x * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.2 + 0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right)\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{5} \cdot {x}^{2}\right) + 1\right)} \]

   3. accelerator-lowering-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. +-commutative N/A

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

   7. *-commutative N/A

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

   8. accelerator-lowering-fma.f64 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{5}, \frac{1}{3}\right)}, 1\right) \]

   9. unpow2 N/A

      \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{5}, \frac{1}{3}\right), 1\right) \]

   10. *-lowering-*.f64 100.0

       \[\leadsto x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.2, 0.3333333333333333\right), 1\right) \]

5. Simplified 100.0%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.2, 0.3333333333333333\right), 1\right)} \]
6. Add Preprocessing

Alternative 5: 99.5% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot \left(x \cdot x\right), 0.3333333333333333, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (* x (* x x)) 0.3333333333333333 x))

C

double code(double x) {
	return fma((x * (x * x)), 0.3333333333333333, x);
}

Julia

function code(x)
	return fma(Float64(x * Float64(x * x)), 0.3333333333333333, x)
end

Wolfram

code[x_] := N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.3333333333333333 + x), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{3} \cdot {x}^{2} + 1\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{3}} + 1\right) \]

   4. unpow2 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{3} + 1\right) \]

   5. associate-*l* N/A

      \[\leadsto x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{3}\right)} + 1\right) \]

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 99.9

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.3333333333333333}, 1\right) \]

5. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot 0.3333333333333333, 1\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) + x \cdot 1} \]

   2. *-rgt-identity N/A

      \[\leadsto x \cdot \left(x \cdot \left(x \cdot \frac{1}{3}\right)\right) + \color{blue}{x} \]

   3. associate-*r* N/A

      \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \frac{1}{3}\right)} + x \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot \frac{1}{3}} + x \]

   5. cube-unmult N/A

      \[\leadsto \color{blue}{{x}^{3}} \cdot \frac{1}{3} + x \]

   6. accelerator-lowering-fma.f64 N/A

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

   7. cube-unmult N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 99.9

      \[\leadsto \mathsf{fma}\left(x \cdot \color{blue}{\left(x \cdot x\right)}, 0.3333333333333333, x\right) \]

7. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), 0.3333333333333333, x\right)} \]
8. Add Preprocessing

Alternative 6: 99.4% accurate, 7.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x, x \cdot 0.3333333333333333, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (fma x (* x 0.3333333333333333) 1.0)))

C

double code(double x) {
	return x * fma(x, (x * 0.3333333333333333), 1.0);
}

Julia

function code(x)
	return Float64(x * fma(x, Float64(x * 0.3333333333333333), 1.0))
end

Wolfram

code[x_] := N[(x * N[(x * N[(x * 0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

   1. *-lowering-*.f64 N/A

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{3} \cdot {x}^{2}\right)} \]

   2. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left(\frac{1}{3} \cdot {x}^{2} + 1\right)} \]

   3. *-commutative N/A

      \[\leadsto x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{3}} + 1\right) \]

   4. unpow2 N/A

      \[\leadsto x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{3} + 1\right) \]

   5. associate-*l* N/A

      \[\leadsto x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{1}{3}\right)} + 1\right) \]

   6. accelerator-lowering-fma.f64 N/A

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

   7. *-lowering-*.f64 99.9

      \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.3333333333333333}, 1\right) \]

5. Simplified 99.9%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot 0.3333333333333333, 1\right)} \]
6. Add Preprocessing

Alternative 7: 98.9% accurate, 125.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in x around 0

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

5. Simplified 99.2%

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

Reproduce

herbie shell --seed 2024194 
(FPCore (x)
  :name "Rust f64::atanh"
  :precision binary64
  (* 0.5 (log1p (/ (* 2.0 x) (- 1.0 x)))))