VandenBroeck and Keller, Equation (20)

Average Accuracy: 4.4% → 95.7%

Time: 20.0s

Precision: binary64

Cost: 19648

Math

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

↓

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

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 (* -4.0 (/ (log (/ (/ 4.0 PI) f)) 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 -4.0 * (log(((4.0 / ((double) M_PI)) / f)) / ((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 -4.0 * (Math.log(((4.0 / Math.PI) / f)) / 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 -4.0 * (math.log(((4.0 / math.pi) / f)) / 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(-4.0 * Float64(log(Float64(Float64(4.0 / pi) / f)) / 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 = -4.0 * (log(((4.0 / pi) / f)) / 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[(-4.0 * N[(N[Log[N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 4.4%

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

2. Simplified 4.4%

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

   [Start] 4.4	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   *-commutative [=>] 4.4	\[ -\color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \frac{1}{\frac{\pi}{4}}} \]
   distribute-rgt-neg-in [=>] 4.4	\[ \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right)} \]

3. Taylor expanded in f around 0 95.8%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi}} \]

4. Taylor expanded in f around inf 95.8%

   \[\leadsto -4 \cdot \color{blue}{\frac{\log \left(\frac{1}{f}\right) + \log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}{\pi}} \]

5. Simplified 95.7%

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

   [Start] 95.8	\[ -4 \cdot \frac{\log \left(\frac{1}{f}\right) + \log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}{\pi} \]
   +-commutative [=>] 95.8	\[ -4 \cdot \frac{\color{blue}{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + \log \left(\frac{1}{f}\right)}}{\pi} \]
   distribute-rgt-out-- [=>] 95.8	\[ -4 \cdot \frac{\log \left(\frac{2}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) + \log \left(\frac{1}{f}\right)}{\pi} \]
   metadata-eval [=>] 95.8	\[ -4 \cdot \frac{\log \left(\frac{2}{\pi \cdot \color{blue}{0.5}}\right) + \log \left(\frac{1}{f}\right)}{\pi} \]
   *-commutative [=>] 95.8	\[ -4 \cdot \frac{\log \left(\frac{2}{\color{blue}{0.5 \cdot \pi}}\right) + \log \left(\frac{1}{f}\right)}{\pi} \]
   associate-/r* [=>] 95.8	\[ -4 \cdot \frac{\log \color{blue}{\left(\frac{\frac{2}{0.5}}{\pi}\right)} + \log \left(\frac{1}{f}\right)}{\pi} \]
   metadata-eval [=>] 95.8	\[ -4 \cdot \frac{\log \left(\frac{\color{blue}{4}}{\pi}\right) + \log \left(\frac{1}{f}\right)}{\pi} \]
   metadata-eval [<=] 95.8	\[ -4 \cdot \frac{\log \left(\frac{\color{blue}{1 \cdot 4}}{\pi}\right) + \log \left(\frac{1}{f}\right)}{\pi} \]
   associate-*l/ [<=] 95.8	\[ -4 \cdot \frac{\log \color{blue}{\left(\frac{1}{\pi} \cdot 4\right)} + \log \left(\frac{1}{f}\right)}{\pi} \]
   associate-/r/ [<=] 95.8	\[ -4 \cdot \frac{\log \color{blue}{\left(\frac{1}{\frac{\pi}{4}}\right)} + \log \left(\frac{1}{f}\right)}{\pi} \]
   log-rec [=>] 95.8	\[ -4 \cdot \frac{\log \left(\frac{1}{\frac{\pi}{4}}\right) + \color{blue}{\left(-\log f\right)}}{\pi} \]
   sub-neg [<=] 95.8	\[ -4 \cdot \frac{\color{blue}{\log \left(\frac{1}{\frac{\pi}{4}}\right) - \log f}}{\pi} \]
   log-div [<=] 95.7	\[ -4 \cdot \frac{\color{blue}{\log \left(\frac{\frac{1}{\frac{\pi}{4}}}{f}\right)}}{\pi} \]
   associate-/r/ [=>] 95.7	\[ -4 \cdot \frac{\log \left(\frac{\color{blue}{\frac{1}{\pi} \cdot 4}}{f}\right)}{\pi} \]
   associate-*l/ [=>] 95.7	\[ -4 \cdot \frac{\log \left(\frac{\color{blue}{\frac{1 \cdot 4}{\pi}}}{f}\right)}{\pi} \]
   metadata-eval [=>] 95.7	\[ -4 \cdot \frac{\log \left(\frac{\frac{\color{blue}{4}}{\pi}}{f}\right)}{\pi} \]

6. Final simplification 95.7%

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

Alternatives

Alternative 1
Accuracy	28.9%
Cost	19520

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

Alternative 2
Accuracy	13.6%
Cost	6528

\[\frac{-2}{\pi} \]

Reproduce

herbie shell --seed 2023126 
(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)))))))))