VandenBroeck and Keller, Equation (20)

Average Error: 61.4 → 0.6

Time: 20.2s

Precision: binary64

Cost: 38912

Math

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

↓

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

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 (exp (log (* f (* PI 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(exp(log((f * (((double) M_PI) * 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.exp(Math.log((f * (Math.PI * 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.exp(math.log((f * (math.pi * 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(exp(log(Float64(f * Float64(pi * 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(exp(log((f * (pi * 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[Exp[N[Log[N[(f * N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]) / N[(Pi * -0.25), $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. Applied egg-rr 2.6

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

3. Applied egg-rr 0.5

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

4. Applied egg-rr 0.6

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

5. Final simplification 0.6

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

Alternatives

Alternative 1
Error	0.5
Cost	26112

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

Alternative 2
Error	2.6
Cost	19712

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

Alternative 3
Error	2.6
Cost	19648

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

Reproduce

herbie shell --seed 2022291 
(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)))))))))