VandenBroeck and Keller, Equation (20)

Average Accuracy: 3.9% → 96.5%

Time: 23.6s

Precision: binary64

Cost: 71808

Math

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

↓

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

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
    (/
     (+ (exp (* PI (* f 0.25))) (exp (* PI (* f -0.25))))
     (fma (* PI 0.5) f (* (pow PI 3.0) (* 0.005208333333333333 (pow f 3.0))))))
   -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(((exp((((double) M_PI) * (f * 0.25))) + exp((((double) M_PI) * (f * -0.25)))) / fma((((double) M_PI) * 0.5), f, (pow(((double) M_PI), 3.0) * (0.005208333333333333 * pow(f, 3.0)))))) * -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(Float64(log(Float64(Float64(exp(Float64(pi * Float64(f * 0.25))) + exp(Float64(pi * Float64(f * -0.25)))) / fma(Float64(pi * 0.5), f, Float64((pi ^ 3.0) * Float64(0.005208333333333333 * (f ^ 3.0)))))) * -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[(N[Log[N[(N[(N[Exp[N[(Pi * N[(f * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(Pi * N[(f * -0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(Pi * 0.5), $MachinePrecision] * f + N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(0.005208333333333333 * N[Power[f, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -4.0), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 3.9%

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

2. Simplified 3.9%

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

   [Start] 3.9	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   *-commutative [=>] 3.9	\[ -\color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \frac{1}{\frac{\pi}{4}}} \]
   distribute-rgt-neg-in [=>] 3.9	\[ \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right)} \]

3. Taylor expanded in f around 0 96.4%

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

4. Applied egg-rr 96.5%

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

   [Start] 96.4	\[ \log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right) \cdot \frac{-4}{\pi} \]
   associate-*r/ [=>] 96.5	\[ \color{blue}{\frac{\log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{-0.25 \cdot \left(\pi \cdot f\right)}}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right) \cdot -4}{\pi}} \]

5. Final simplification 96.5%

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

Alternatives

Alternative 1
Accuracy	96.5%
Cost	45888

\[\frac{-4 \cdot \log \left(\frac{0}{\pi} + \left(\left(\left(\pi \cdot f\right) \cdot 0.08333333333333333 + e^{\mathsf{log1p}\left(\frac{4}{\pi \cdot f}\right)}\right) + -1\right)\right)}{\pi} \]

Alternative 2
Accuracy	96.5%
Cost	32640

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

Alternative 3
Accuracy	96.0%
Cost	26048

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

Alternative 4
Accuracy	95.9%
Cost	19648

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

Alternative 5
Accuracy	95.9%
Cost	19648

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

Alternative 6
Accuracy	96.0%
Cost	19648

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

Alternative 7
Accuracy	0.0%
Cost	13056

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

Reproduce

herbie shell --seed 2023135 
(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)))))))))