VandenBroeck and Keller, Equation (20)

Average Error: 61.4 → 2.7

Time: 19.6s

Precision: binary64

Cost: 39232

Math

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

↓

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

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
 (*
  (log (+ (/ 0.0 PI) (fma f (* PI 0.08333333333333333) (/ 4.0 (* PI f)))))
  (/ -4.0 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 log(((0.0 / ((double) M_PI)) + fma(f, (((double) M_PI) * 0.08333333333333333), (4.0 / (((double) M_PI) * f))))) * (-4.0 / ((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 Float64(log(Float64(Float64(0.0 / pi) + fma(f, Float64(pi * 0.08333333333333333), Float64(4.0 / Float64(pi * f))))) * Float64(-4.0 / 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[(N[Log[N[(N[(0.0 / Pi), $MachinePrecision] + N[(f * N[(Pi * 0.08333333333333333), $MachinePrecision] + N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $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 2.7

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

4. Applied egg-rr 2.7

   \[\leadsto \color{blue}{\left(\log \left(\frac{e^{\pi \cdot \left(f \cdot 0.25\right)} + e^{\pi \cdot \left(-0.25 \cdot f\right)}}{\mathsf{fma}\left(\pi \cdot 0.5, f, {\pi}^{3} \cdot \left(0.005208333333333333 \cdot {f}^{3}\right)\right)}\right) \cdot 1\right)} \cdot \frac{-4}{\pi} \]

5. Taylor expanded in f around 0 2.7

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

6. Simplified 2.7

   \[\leadsto \left(\log \color{blue}{\left(\frac{0}{\pi} + \mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{4}{f \cdot \pi}\right)\right)} \cdot 1\right) \cdot \frac{-4}{\pi} \]
   Proof

   [Start] 2.7	\[ \left(\log \left(2 \cdot \frac{-0.25 \cdot \pi + 0.25 \cdot \pi}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot 1\right) \cdot \frac{-4}{\pi} \]
   associate-*r/ [=>] 2.7	\[ \left(\log \left(\color{blue}{\frac{2 \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}{\pi}} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot 1\right) \cdot \frac{-4}{\pi} \]
   *-commutative [=>] 2.7	\[ \left(\log \left(\frac{\color{blue}{\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right) \cdot 2}}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot 1\right) \cdot \frac{-4}{\pi} \]
   distribute-rgt-out [=>] 2.7	\[ \left(\log \left(\frac{\color{blue}{\left(\pi \cdot \left(-0.25 + 0.25\right)\right)} \cdot 2}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot 1\right) \cdot \frac{-4}{\pi} \]
   metadata-eval [=>] 2.7	\[ \left(\log \left(\frac{\left(\pi \cdot \color{blue}{0}\right) \cdot 2}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot 1\right) \cdot \frac{-4}{\pi} \]
   mul0-rgt [=>] 2.7	\[ \left(\log \left(\frac{\color{blue}{0} \cdot 2}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot 1\right) \cdot \frac{-4}{\pi} \]
   metadata-eval [=>] 2.7	\[ \left(\log \left(\frac{\color{blue}{0}}{\pi} + \left(f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot 1\right) \cdot \frac{-4}{\pi} \]
   fma-def [=>] 2.7	\[ \left(\log \left(\frac{0}{\pi} + \color{blue}{\mathsf{fma}\left(f, 0.125 \cdot \pi - 0.041666666666666664 \cdot \pi, 4 \cdot \frac{1}{f \cdot \pi}\right)}\right) \cdot 1\right) \cdot \frac{-4}{\pi} \]
   distribute-rgt-out-- [=>] 2.7	\[ \left(\log \left(\frac{0}{\pi} + \mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.125 - 0.041666666666666664\right)}, 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot 1\right) \cdot \frac{-4}{\pi} \]
   metadata-eval [=>] 2.7	\[ \left(\log \left(\frac{0}{\pi} + \mathsf{fma}\left(f, \pi \cdot \color{blue}{0.08333333333333333}, 4 \cdot \frac{1}{f \cdot \pi}\right)\right) \cdot 1\right) \cdot \frac{-4}{\pi} \]
   associate-*r/ [=>] 2.7	\[ \left(\log \left(\frac{0}{\pi} + \mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \color{blue}{\frac{4 \cdot 1}{f \cdot \pi}}\right)\right) \cdot 1\right) \cdot \frac{-4}{\pi} \]
   metadata-eval [=>] 2.7	\[ \left(\log \left(\frac{0}{\pi} + \mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\color{blue}{4}}{f \cdot \pi}\right)\right) \cdot 1\right) \cdot \frac{-4}{\pi} \]

7. Final simplification 2.7

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

Alternatives

Alternative 1
Error	2.9
Cost	26048

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

Alternative 2
Error	3.0
Cost	19648

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

Alternative 3
Error	3.0
Cost	19648

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

Alternative 4
Error	43.9
Cost	13248

\[\frac{-\log \left(f \cdot 0.5\right)}{\pi \cdot -0.25} \]

Alternative 5
Error	53.0
Cost	6720

\[4 \cdot \log \left(f \cdot 0.5\right) \]

Reproduce

herbie shell --seed 2023059 
(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)))))))))