VandenBroeck and Keller, Equation (20)

Average Error: 61.5 → 0.6

Time: 15.8s

Precision: binary64

Cost: 26048

Math

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

↓

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

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 (* 0.25 (* f PI)))) 4.0) PI))

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((0.25 * (f * ((double) M_PI))))) * 4.0) / ((double) M_PI);
}

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((0.25 * (f * Math.PI)))) * 4.0) / Math.PI;
}

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((0.25 * (f * math.pi)))) * 4.0) / math.pi

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(0.25 * Float64(f * pi)))) * 4.0) / pi)
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((0.25 * (f * pi)))) * 4.0) / pi;
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[(0.25 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * 4.0), $MachinePrecision] / Pi), $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. Applied egg-rr 2.0

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

3. Applied egg-rr 14.3

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

4. Applied egg-rr 14.3

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

5. Applied egg-rr 0.6

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

6. Final simplification 0.6

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

Alternatives

Alternative 1
Error	2.8
Cost	19648

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

Alternative 2
Error	2.7
Cost	19648

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

Alternative 3
Error	60.7
Cost	64

\[0 \]

Reproduce

herbie shell --seed 2022325 
(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)))))))))