VandenBroeck and Keller, Equation (20)

Average Accuracy: 4.0% → 96.3%

Time: 22.1s

Precision: binary64

Cost: 136320

Math

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

↓

\[\log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{\pi \cdot -0.25}\right)}^{f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {\pi}^{7} \cdot \left(2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}\right)\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

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
   (/
    (+ (pow (exp f) (/ PI 4.0)) (pow (exp (* PI -0.25)) f))
    (fma
     (pow f 5.0)
     (* (pow PI 5.0) 1.6276041666666666e-5)
     (fma
      (* PI 0.5)
      f
      (fma
       (pow f 3.0)
       (* (pow PI 3.0) 0.005208333333333333)
       (* (pow PI 7.0) (* 2.422030009920635e-8 (pow f 7.0))))))))
  (/ -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(((pow(exp(f), (((double) M_PI) / 4.0)) + pow(exp((((double) M_PI) * -0.25)), f)) / fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), fma((((double) M_PI) * 0.5), f, fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), (pow(((double) M_PI), 7.0) * (2.422030009920635e-8 * pow(f, 7.0)))))))) * (-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(log(Float64(Float64((exp(f) ^ Float64(pi / 4.0)) + (exp(Float64(pi * -0.25)) ^ f)) / fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), fma(Float64(pi * 0.5), f, fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), Float64((pi ^ 7.0) * Float64(2.422030009920635e-8 * (f ^ 7.0)))))))) * 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[(N[Power[N[Exp[f], $MachinePrecision], N[(Pi / 4.0), $MachinePrecision]], $MachinePrecision] + N[Power[N[Exp[N[(Pi * -0.25), $MachinePrecision]], $MachinePrecision], f], $MachinePrecision]), $MachinePrecision] / N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(N[(Pi * 0.5), $MachinePrecision] * f + N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(N[Power[Pi, 7.0], $MachinePrecision] * N[(2.422030009920635e-8 * N[Power[f, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 4.0%

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

2. Simplified 3.9%

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

   [Start] 4.0	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   *-commutative [=>] 4.0	\[ -\color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \frac{1}{\frac{\pi}{4}}} \]
   distribute-rgt-neg-in [=>] 4.0	\[ \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right)} \]

3. Taylor expanded in f around 0 96.3%

   \[\leadsto \log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{-0.25 \cdot \pi}\right)}^{f}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

4. Simplified 96.3%

   \[\leadsto \log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{-0.25 \cdot \pi}\right)}^{f}}{\color{blue}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {\pi}^{7} \cdot \left(2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}\right)\right)\right)\right)}}\right) \cdot \frac{-4}{\pi} \]
   Proof

   [Start] 96.3	\[ \log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{-0.25 \cdot \pi}\right)}^{f}}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   fma-def [=>] 96.3	\[ \log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{-0.25 \cdot \pi}\right)}^{f}}{\color{blue}{\mathsf{fma}\left({f}^{5}, 8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}, \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \cdot \frac{-4}{\pi} \]
   distribute-rgt-out-- [=>] 96.3	\[ \log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{-0.25 \cdot \pi}\right)}^{f}}{\mathsf{fma}\left({f}^{5}, \color{blue}{{\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)}, \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   metadata-eval [=>] 96.3	\[ \log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{-0.25 \cdot \pi}\right)}^{f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

5. Final simplification 96.3%

   \[\leadsto \log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{\pi \cdot -0.25}\right)}^{f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {\pi}^{7} \cdot \left(2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}\right)\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

Alternatives

Alternative 1
Accuracy	96.2%
Cost	110400

\[\frac{-4}{\pi} \cdot \log \left(\frac{{\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)} + {\left(e^{\pi \cdot -0.25}\right)}^{f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \pi \cdot \left(f \cdot 0.5\right)\right)\right)}\right) \]

Alternative 2
Accuracy	96.1%
Cost	52672

\[\frac{-4}{\pi} \cdot \left(\log \left(\frac{4}{\pi}\right) - \log f\right) + \pi \cdot \left(\left(f \cdot f\right) \cdot -0.08333333333333333 + 0.125 \cdot \left({f}^{4} \cdot \left({\pi}^{2} \cdot 0.006944444444444444\right)\right)\right) \]

Alternative 3
Accuracy	96.0%
Cost	26496

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

Alternative 4
Accuracy	95.6%
Cost	26048

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

Alternative 5
Accuracy	95.4%
Cost	19648

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

Alternative 6
Accuracy	95.5%
Cost	19648

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

Alternative 7
Accuracy	95.6%
Cost	19648

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

Alternative 8
Accuracy	13.6%
Cost	6528

\[\frac{-2}{\pi} \]

Reproduce

herbie shell --seed 2023140 
(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)))))))))