VandenBroeck and Keller, Equation (20)

Average Error: 61.5 → 1.6

Time: 39.4s

Precision: binary64

Cost: 64832

Math

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

↓

\[\frac{-4}{\pi} \cdot \log \left({\left({\left(\frac{1 + {\left(\sqrt{e^{f}}\right)}^{\pi}}{\mathsf{expm1}\left(\left(f \cdot 0.5\right) \cdot \pi\right)}\right)}^{0.3333333333333333}\right)}^{3}\right) \]

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
 (*
  (/ -4.0 PI)
  (log
   (pow
    (pow
     (/ (+ 1.0 (pow (sqrt (exp f)) PI)) (expm1 (* (* f 0.5) PI)))
     0.3333333333333333)
    3.0))))

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 (-4.0 / ((double) M_PI)) * log(pow(pow(((1.0 + pow(sqrt(exp(f)), ((double) M_PI))) / expm1(((f * 0.5) * ((double) M_PI)))), 0.3333333333333333), 3.0));
}

Java

public static double code(double f) {
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((Math.exp(((Math.PI / 4.0) * f)) + Math.exp(-((Math.PI / 4.0) * f))) / (Math.exp(((Math.PI / 4.0) * f)) - Math.exp(-((Math.PI / 4.0) * f))))));
}

↓

public static double code(double f) {
	return (-4.0 / Math.PI) * Math.log(Math.pow(Math.pow(((1.0 + Math.pow(Math.sqrt(Math.exp(f)), Math.PI)) / Math.expm1(((f * 0.5) * Math.PI))), 0.3333333333333333), 3.0));
}

Python

def code(f):
	return -((1.0 / (math.pi / 4.0)) * math.log(((math.exp(((math.pi / 4.0) * f)) + math.exp(-((math.pi / 4.0) * f))) / (math.exp(((math.pi / 4.0) * f)) - math.exp(-((math.pi / 4.0) * f))))))

↓

def code(f):
	return (-4.0 / math.pi) * math.log(math.pow(math.pow(((1.0 + math.pow(math.sqrt(math.exp(f)), math.pi)) / math.expm1(((f * 0.5) * math.pi))), 0.3333333333333333), 3.0))

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 Float64(Float64(-4.0 / pi) * log(((Float64(Float64(1.0 + (sqrt(exp(f)) ^ pi)) / expm1(Float64(Float64(f * 0.5) * pi))) ^ 0.3333333333333333) ^ 3.0)))
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[(N[(-4.0 / Pi), $MachinePrecision] * N[Log[N[Power[N[Power[N[(N[(1.0 + N[Power[N[Sqrt[N[Exp[f], $MachinePrecision]], $MachinePrecision], Pi], $MachinePrecision]), $MachinePrecision] / N[(Exp[N[(N[(f * 0.5), $MachinePrecision] * Pi), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], 0.3333333333333333], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 61.5

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

2. Simplified 1.8

   \[\leadsto \color{blue}{\frac{-4}{\pi} \cdot \log \left(\frac{e^{\left(0.25 \cdot f\right) \cdot \left(\pi + \pi\right)} + 1}{\mathsf{expm1}\left(\left(0.25 \cdot f\right) \cdot \left(\pi + \pi\right)\right)}\right)} \]
   Proof

3. Applied egg-rr 32.1

   \[\leadsto \frac{-4}{\pi} \cdot \log \color{blue}{\left(\frac{\frac{-1 - {\left(e^{0.25 \cdot f}\right)}^{\left(\pi + \pi\right)}}{\sqrt[3]{-\mathsf{expm1}\left(\left(0.25 \cdot f\right) \cdot \left(\pi + \pi\right)\right)}}}{\sqrt[3]{{\left(\mathsf{expm1}\left(\left(0.25 \cdot f\right) \cdot \left(\pi + \pi\right)\right)\right)}^{2}}}\right)} \]

4. Simplified 32.1

   \[\leadsto \frac{-4}{\pi} \cdot \log \color{blue}{\left(\frac{\frac{-1 - {\left(e^{0.5}\right)}^{\left(f \cdot \pi\right)}}{\sqrt[3]{-\mathsf{expm1}\left(f \cdot \left(0.5 \cdot \pi\right)\right)}}}{\sqrt[3]{{\left(\mathsf{expm1}\left(f \cdot \left(0.5 \cdot \pi\right)\right)\right)}^{2}}}\right)} \]
   Proof

5. Applied egg-rr 1.8

   \[\leadsto \frac{-4}{\pi} \cdot \log \color{blue}{\left({\left(\sqrt[3]{\frac{1 + {\left(\sqrt{e}\right)}^{\left(f \cdot \pi\right)}}{\mathsf{expm1}\left(\left(f \cdot 0.5\right) \cdot \pi\right)}}\right)}^{3}\right)} \]

6. Applied egg-rr 1.6

   \[\leadsto \frac{-4}{\pi} \cdot \log \left({\color{blue}{\left({\left(\frac{1 + {\left(\sqrt{e^{f}}\right)}^{\pi}}{\mathsf{expm1}\left(\left(f \cdot 0.5\right) \cdot \pi\right)}\right)}^{0.3333333333333333}\right)}}^{3}\right) \]

Alternatives

Alternative 1
Error	1.7
Cost	51904

\[\frac{-4 \cdot \log \left(\frac{1 + {\left(\sqrt{e^{f}}\right)}^{\pi}}{\mathsf{expm1}\left(\left(f \cdot 0.5\right) \cdot \pi\right)}\right)}{\pi} \]

Alternative 2
Error	1.8
Cost	45632

\[\frac{-4}{\pi} \cdot \log \left(\frac{{\left(e^{0.5}\right)}^{\left(f \cdot \pi\right)} + 1}{\mathsf{expm1}\left(f \cdot \left(0.5 \cdot \pi\right)\right)}\right) \]

Alternative 3
Error	42.3
Cost	39556

\[\begin{array}{l} t_0 := \sqrt[3]{\frac{1}{\pi}}\\ t_1 := e^{0.5 \cdot \left(f \cdot \pi\right)}\\ \mathbf{if}\;f \leq 9 \cdot 10^{-17}:\\ \;\;\;\;t_0 \cdot \left({t_0}^{2} \cdot \left(-4 \cdot \log \left(\frac{2}{f}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-4}{\pi} \cdot \log \left(\frac{1 + t_1}{t_1 - 1}\right)\\ \end{array} \]

Alternative 4
Error	43.7
Cost	19968

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

Alternative 5
Error	43.8
Cost	13184

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

Alternative 6
Error	43.8
Cost	13184

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

Reproduce

herbie shell --seed 2023010 
(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)))))))))