VandenBroeck and Keller, Equation (20)

Average Error: 61.4 → 2.4

Time: 20.3s

Precision: binary64

Cost: 71744

Math

\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

↓

\[\mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, \mathsf{fma}\left(-2, \frac{f \cdot f}{\pi} \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right), f \cdot \frac{0}{\pi}\right)\right) \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (*
   (/ 1.0 (/ PI 4.0))
   (log
    (/
     (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f))))
     (- (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))))))))

↓

(FPCore (f)
 :precision binary64
 (fma
  -4.0
  (/ (- (log (/ 4.0 PI)) (log f)) PI)
  (fma
   -2.0
   (* (/ (* f f) PI) (fma 0.5 (* (pow PI 2.0) 0.08333333333333333) 0.0))
   (* f (/ 0.0 PI)))))

C

double code(double f) {
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((exp(((((double) M_PI) / 4.0) * f)) + exp(-((((double) M_PI) / 4.0) * f))) / (exp(((((double) M_PI) / 4.0) * f)) - exp(-((((double) M_PI) / 4.0) * f))))));
}

↓

double code(double f) {
	return fma(-4.0, ((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI)), fma(-2.0, (((f * f) / ((double) M_PI)) * fma(0.5, (pow(((double) M_PI), 2.0) * 0.08333333333333333), 0.0)), (f * (0.0 / ((double) M_PI)))));
}

Julia

function code(f)
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(exp(Float64(Float64(pi / 4.0) * f)) + exp(Float64(-Float64(Float64(pi / 4.0) * f)))) / Float64(exp(Float64(Float64(pi / 4.0) * f)) - exp(Float64(-Float64(Float64(pi / 4.0) * f))))))))
end

↓

function code(f)
	return fma(-4.0, Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi), fma(-2.0, Float64(Float64(Float64(f * f) / pi) * fma(0.5, Float64((pi ^ 2.0) * 0.08333333333333333), 0.0)), Float64(f * Float64(0.0 / pi))))
end

Wolfram

code[f_] := (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision] + N[Exp[(-N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision])], $MachinePrecision]), $MachinePrecision] / N[(N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision] - N[Exp[(-N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])

↓

code[f_] := N[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] + N[(-2.0 * N[(N[(N[(f * f), $MachinePrecision] / Pi), $MachinePrecision] * N[(0.5 * N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision] + N[(f * N[(0.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.4

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]

2. Simplified 61.4

   \[\leadsto \color{blue}{\log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{e^{\frac{\pi \cdot f}{4}} - e^{-0.25 \cdot \left(\pi \cdot f\right)}}\right) \cdot \frac{-4}{\pi}} \]
   Proof

   [Start] 61.4	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   *-commutative [=>] 61.4	\[ -\color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \frac{1}{\frac{\pi}{4}}} \]
   distribute-rgt-neg-in [=>] 61.4	\[ \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right)} \]

3. Taylor expanded in f around 0 2.5

   \[\leadsto \log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{\color{blue}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}}\right) \cdot \frac{-4}{\pi} \]

4. Applied egg-rr 3.2

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\log \left(\frac{e^{\pi \cdot \left(f \cdot 0.25\right)} + e^{\pi \cdot \left(-0.25 \cdot f\right)}}{\mathsf{fma}\left(\pi \cdot 0.5, f, {\pi}^{3} \cdot \left(0.005208333333333333 \cdot {f}^{3}\right)\right)}\right)\right)\right)} \cdot \frac{-4}{\pi} \]

5. Taylor expanded in f around 0 2.4

   \[\leadsto \color{blue}{-4 \cdot \frac{-1 \cdot \log f + \log \left(\frac{4}{\pi}\right)}{\pi} + \left(-2 \cdot \frac{{f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)}{\pi} + -2 \cdot \frac{\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right) \cdot f}{\pi}\right)} \]

6. Simplified 2.4

   \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, \mathsf{fma}\left(-2, \frac{f \cdot f}{\pi} \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right), \frac{0}{\pi} \cdot f\right)\right)} \]
   Proof

   [Start] 2.4	\[ -4 \cdot \frac{-1 \cdot \log f + \log \left(\frac{4}{\pi}\right)}{\pi} + \left(-2 \cdot \frac{{f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)}{\pi} + -2 \cdot \frac{\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right) \cdot f}{\pi}\right) \]
   fma-def [=>] 2.4	\[ \color{blue}{\mathsf{fma}\left(-4, \frac{-1 \cdot \log f + \log \left(\frac{4}{\pi}\right)}{\pi}, -2 \cdot \frac{{f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)}{\pi} + -2 \cdot \frac{\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right) \cdot f}{\pi}\right)} \]
   +-commutative [<=] 2.4	\[ \mathsf{fma}\left(-4, \frac{\color{blue}{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}}{\pi}, -2 \cdot \frac{{f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)}{\pi} + -2 \cdot \frac{\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right) \cdot f}{\pi}\right) \]
   mul-1-neg [=>] 2.4	\[ \mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(-\log f\right)}}{\pi}, -2 \cdot \frac{{f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)}{\pi} + -2 \cdot \frac{\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right) \cdot f}{\pi}\right) \]
   unsub-neg [=>] 2.4	\[ \mathsf{fma}\left(-4, \frac{\color{blue}{\log \left(\frac{4}{\pi}\right) - \log f}}{\pi}, -2 \cdot \frac{{f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)}{\pi} + -2 \cdot \frac{\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right) \cdot f}{\pi}\right) \]
   fma-def [=>] 2.4	\[ \mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, \color{blue}{\mathsf{fma}\left(-2, \frac{{f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)}{\pi}, -2 \cdot \frac{\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right) \cdot f}{\pi}\right)}\right) \]

7. Final simplification 2.4

   \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, \mathsf{fma}\left(-2, \frac{f \cdot f}{\pi} \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right), f \cdot \frac{0}{\pi}\right)\right) \]

Alternatives

Alternative 1
Error	2.5
Cost	46016

\[\left(\left(\log \left(\frac{4}{\pi}\right) - \log f\right) + 0.5 \cdot \left(f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]

Alternative 2
Error	2.9
Cost	26048

\[\left(\log \left(\frac{4}{\pi}\right) - \log f\right) \cdot \frac{-4}{\pi} \]

Alternative 3
Error	2.8
Cost	26048

\[\frac{-4 \cdot \left(\log \left(\frac{4}{\pi}\right) - \log f\right)}{\pi} \]

Alternative 4
Error	44.0
Cost	19648

\[\frac{-4}{\pi} \cdot \log \left(\frac{\pi}{\frac{f}{0.25}}\right) \]

Alternative 5
Error	2.9
Cost	19648

\[\frac{4}{\pi} \cdot \log \left(f \cdot \left(\pi \cdot 0.25\right)\right) \]

Alternative 6
Error	45.5
Cost	19520

\[\frac{4}{\pi} \cdot \log \left(\pi \cdot f\right) \]

Alternative 7
Error	64.0
Cost	13056

\[\frac{-4}{\pi} \cdot \mathsf{log1p}\left(-2\right) \]

Reproduce

herbie shell --seed 2023053 
(FPCore (f)
  :name "VandenBroeck and Keller, Equation (20)"
  :precision binary64
  (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))) (- (exp (* (/ PI 4.0) f)) (exp (- (* (/ PI 4.0) f)))))))))