VandenBroeck and Keller, Equation (20)

Average Accuracy: 4.0% → 99.0%

Time: 23.3s

Precision: binary64

Cost: 26112

Math

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

↓

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

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 (* f 0.25)))) (* 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) * (f * 0.25)))) / (((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 * (f * 0.25)))) / (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 * (f * 0.25)))) / (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(f * 0.25)))) / Float64(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 * (f * 0.25)))) / (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[Log[N[Tanh[N[(Pi * N[(f * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(Pi * -0.25), $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. Applied egg-rr 96.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{e^{\log \log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{2 \cdot \sinh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}\right)}} \]
   Proof

   [Start] 4.0	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   add-exp-log [=>] 4.0	\[ -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{e^{\log \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)}} \]
   cosh-undef [=>] 4.0	\[ -\frac{1}{\frac{\pi}{4}} \cdot e^{\log \log \left(\frac{\color{blue}{2 \cdot \cosh \left(\frac{\pi}{4} \cdot f\right)}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)} \]
   div-inv [=>] 4.0	\[ -\frac{1}{\frac{\pi}{4}} \cdot e^{\log \log \left(\frac{2 \cdot \cosh \left(\color{blue}{\left(\pi \cdot \frac{1}{4}\right)} \cdot f\right)}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)} \]
   metadata-eval [=>] 4.0	\[ -\frac{1}{\frac{\pi}{4}} \cdot e^{\log \log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot \color{blue}{0.25}\right) \cdot f\right)}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)} \]
   sinh-undef [=>] 96.0	\[ -\frac{1}{\frac{\pi}{4}} \cdot e^{\log \log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{\color{blue}{2 \cdot \sinh \left(\frac{\pi}{4} \cdot f\right)}}\right)} \]

3. Applied egg-rr 97.7%

   \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\log \tanh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\pi \cdot -0.25}\right)} - 1\right)} \]
   Proof

   [Start] 96.0	\[ -\frac{1}{\frac{\pi}{4}} \cdot e^{\log \log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{2 \cdot \sinh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}\right)} \]
   expm1-log1p-u [=>] 95.6	\[ -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot e^{\log \log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{2 \cdot \sinh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}\right)}\right)\right)} \]
   expm1-udef [=>] 95.6	\[ -\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot e^{\log \log \left(\frac{2 \cdot \cosh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}{2 \cdot \sinh \left(\left(\pi \cdot 0.25\right) \cdot f\right)}\right)}\right)} - 1\right)} \]

4. Simplified 99.0%

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

   [Start] 97.7	\[ -\left(e^{\mathsf{log1p}\left(\frac{\log \tanh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\pi \cdot -0.25}\right)} - 1\right) \]
   expm1-def [=>] 97.7	\[ -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \tanh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\pi \cdot -0.25}\right)\right)} \]
   expm1-log1p [=>] 99.0	\[ -\color{blue}{\frac{\log \tanh \left(\pi \cdot \left(0.25 \cdot f\right)\right)}{\pi \cdot -0.25}} \]
   *-commutative [=>] 99.0	\[ -\frac{\log \tanh \left(\pi \cdot \color{blue}{\left(f \cdot 0.25\right)}\right)}{\pi \cdot -0.25} \]

5. Final simplification 99.0%

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

Alternatives

Alternative 1
Accuracy	96.1%
Cost	26048

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

Alternative 2
Accuracy	96.0%
Cost	19712

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

Alternative 3
Accuracy	0.0%
Cost	19648

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

Alternative 4
Accuracy	95.9%
Cost	19648

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

Alternative 5
Accuracy	95.8%
Cost	19648

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

Reproduce

herbie shell --seed 2023136 
(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)))))))))