VandenBroeck and Keller, Equation (20)

Average Error: 61.4 → 2.2

Time: 14.9s

Precision: binary64

Math

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

↓

\[\mathsf{fma}\left(-4, \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, -2 \cdot \left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right)\right), 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

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 (/ 2.0 (* PI 0.5))) (log f)) PI)
  (*
   -2.0
   (fma
    (/
     (fma
      (* PI 0.5)
      (fma
       0.0625
       (/ PI 0.5)
       (* -2.0 (* (/ (pow PI 3.0) (pow PI 2.0)) 0.020833333333333332)))
      0.0)
     PI)
    (* f f)
    0.0))))

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((2.0 / (((double) M_PI) * 0.5))) - log(f)) / ((double) M_PI)), (-2.0 * fma((fma((((double) M_PI) * 0.5), fma(0.0625, (((double) M_PI) / 0.5), (-2.0 * ((pow(((double) M_PI), 3.0) / pow(((double) M_PI), 2.0)) * 0.020833333333333332))), 0.0) / ((double) M_PI)), (f * f), 0.0)));
}

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(2.0 / Float64(pi * 0.5))) - log(f)) / pi), Float64(-2.0 * fma(Float64(fma(Float64(pi * 0.5), fma(0.0625, Float64(pi / 0.5), Float64(-2.0 * Float64(Float64((pi ^ 3.0) / (pi ^ 2.0)) * 0.020833333333333332))), 0.0) / pi), Float64(f * f), 0.0)))
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[(2.0 / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] + N[(-2.0 * N[(N[(N[(N[(Pi * 0.5), $MachinePrecision] * N[(0.0625 * N[(Pi / 0.5), $MachinePrecision] + N[(-2.0 * N[(N[(N[Power[Pi, 3.0], $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision] / Pi), $MachinePrecision] * N[(f * f), $MachinePrecision] + 0.0), $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

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

3. Taylor expanded in f around 0 2.2

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{\left(-0.25 \cdot \left({\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2} \cdot {\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2}\right) + \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right) \cdot {f}^{2}}{\pi} + -2 \cdot \frac{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot \left(f \cdot \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)\right)}{\pi}\right)} \]

4. Simplified 2.2

   \[\leadsto \color{blue}{\mathsf{fma}\left(-4, \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, 1 \cdot \frac{\pi}{0.5}, \left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right) \cdot -2\right), 0\right)}{\pi}, f \cdot f, 0\right)\right)} \]

5. Final simplification 2.2

   \[\leadsto \mathsf{fma}\left(-4, \frac{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f}{\pi}, -2 \cdot \mathsf{fma}\left(\frac{\mathsf{fma}\left(\pi \cdot 0.5, \mathsf{fma}\left(0.0625, \frac{\pi}{0.5}, -2 \cdot \left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332\right)\right), 0\right)}{\pi}, f \cdot f, 0\right)\right) \]

Reproduce

herbie shell --seed 2022206 
(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)))))))))