VandenBroeck and Keller, Equation (20)

Average Accuracy: 4.2% → 99.1%

Time: 25.3s

Precision: binary64

Cost: 26048

Math

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

↓

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

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 (tanh (* PI (* 0.25 f)))) PI) 0.25))

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(tanh((((double) M_PI) * (0.25 * f)))) / ((double) M_PI)) / 0.25;
}

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 (Math.log(Math.tanh((Math.PI * (0.25 * f)))) / Math.PI) / 0.25;
}

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 (math.log(math.tanh((math.pi * (0.25 * f)))) / math.pi) / 0.25

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(tanh(Float64(pi * Float64(0.25 * f)))) / pi) / 0.25)
end

MATLAB

function tmp = code(f)
	tmp = -((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))))));
end

↓

function tmp = code(f)
	tmp = (log(tanh((pi * (0.25 * f)))) / pi) / 0.25;
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[Tanh[N[(Pi * N[(0.25 * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] / 0.25), $MachinePrecision]

Derivation

1. Initial program 4.2%

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

2. Simplified 4.2%

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

   [Start] 4.2	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   distribute-lft-neg-in [=>] 4.2	\[ \color{blue}{\left(-\frac{1}{\frac{\pi}{4}}\right) \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)} \]
   exp-prod [=>] 4.1	\[ \left(-\frac{1}{\frac{\pi}{4}}\right) \cdot \log \left(\frac{\color{blue}{{\left(e^{\frac{\pi}{4}}\right)}^{f}} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   distribute-rgt-neg-in [=>] 4.1	\[ \left(-\frac{1}{\frac{\pi}{4}}\right) \cdot \log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{f} + e^{\color{blue}{\frac{\pi}{4} \cdot \left(-f\right)}}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   exp-prod [=>] 4.2	\[ \left(-\frac{1}{\frac{\pi}{4}}\right) \cdot \log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\color{blue}{{\left(e^{\frac{\pi}{4}}\right)}^{f}} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   distribute-rgt-neg-in [=>] 4.2	\[ \left(-\frac{1}{\frac{\pi}{4}}\right) \cdot \log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{{\left(e^{\frac{\pi}{4}}\right)}^{f} - e^{\color{blue}{\frac{\pi}{4} \cdot \left(-f\right)}}}\right) \]

3. Applied egg-rr 98.9%

   \[\leadsto \left(-\frac{1}{\frac{\pi}{4}}\right) \cdot \color{blue}{\left(\left(-\log \tanh \left(\pi \cdot \left(0.25 \cdot f\right)\right)\right) + 0\right)} \]

4. Simplified 98.9%

   \[\leadsto \left(-\frac{1}{\frac{\pi}{4}}\right) \cdot \color{blue}{\left(-\log \tanh \left(\pi \cdot \left(0.25 \cdot f\right)\right)\right)} \]
   Proof

   [Start] 98.9	\[ \left(-\frac{1}{\frac{\pi}{4}}\right) \cdot \left(\left(-\log \tanh \left(\pi \cdot \left(0.25 \cdot f\right)\right)\right) + 0\right) \]
   +-rgt-identity [=>] 98.9	\[ \left(-\frac{1}{\frac{\pi}{4}}\right) \cdot \color{blue}{\left(-\log \tanh \left(\pi \cdot \left(0.25 \cdot f\right)\right)\right)} \]

5. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{\frac{\log \tanh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\pi}}{0.25}} \]

6. Final simplification 99.1%

   \[\leadsto \frac{\frac{\log \tanh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\pi}}{0.25} \]

Alternatives

Alternative 1
Accuracy	95.7%
Cost	26048

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

Alternative 2
Accuracy	95.5%
Cost	19904

\[\frac{-1}{\frac{\pi}{4}} \cdot \log \left(\frac{2}{f \cdot \left(\pi \cdot 0.5\right)}\right) \]

Alternative 3
Accuracy	28.8%
Cost	19520

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

Alternative 4
Accuracy	0.0%
Cost	13056

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

Reproduce

herbie shell --seed 2023122 
(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)))))))))