VandenBroeck and Keller, Equation (20)

Average Error: 61.5 → 2.0

Time: 19.4s

Precision: binary64

Cost: 26368

Math

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

↓

\[\frac{\frac{\log \left(0.08333333333333333 \cdot \left(\pi \cdot f\right) + \frac{4}{\pi \cdot f}\right)}{\pi}}{-0.25} \]

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 (+ (* 0.08333333333333333 (* PI f)) (/ 4.0 (* PI 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(((0.08333333333333333 * (((double) M_PI) * f)) + (4.0 / (((double) M_PI) * 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(((0.08333333333333333 * (Math.PI * f)) + (4.0 / (Math.PI * 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(((0.08333333333333333 * (math.pi * f)) + (4.0 / (math.pi * 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(Float64(Float64(0.08333333333333333 * Float64(pi * f)) + Float64(4.0 / Float64(pi * 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(((0.08333333333333333 * (pi * f)) + (4.0 / (pi * 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[(N[(0.08333333333333333 * N[(Pi * f), $MachinePrecision]), $MachinePrecision] + N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision] / -0.25), $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. Simplified 61.5

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

   [Start] 61.5	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   rational.json-simplify-10 [=>] 61.5	\[ \color{blue}{\frac{\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)}{-1}} \]
   rational.json-simplify-49 [=>] 61.5	\[ \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \frac{\frac{1}{\frac{\pi}{4}}}{-1}} \]

3. Taylor expanded in f around 0 2.1

   \[\leadsto \log \color{blue}{\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(2 \cdot \frac{1}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right)\right)\right)} \cdot \frac{-4}{\pi} \]

4. Simplified 2.1

   \[\leadsto \log \color{blue}{\left(2 \cdot \frac{\frac{2}{f}}{\pi} + f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{\pi \cdot 0.5} - 2 \cdot \frac{{\pi}^{3} \cdot 0.005208333333333333}{{\left(\pi \cdot 0.5\right)}^{2}}\right)\right)} \cdot \frac{-4}{\pi} \]
   Proof

   [Start] 2.1

   \[ \log \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(2 \cdot \frac{1}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right)\right)\right)\right) \cdot \frac{-4}{\pi} \]

   rational.json-simplify-41 [=>] 2.1

   \[ \log \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \color{blue}{\left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right) + 2 \cdot \frac{1}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f}\right)\right)}\right) \cdot \frac{-4}{\pi} \]

5. Taylor expanded in f around 0 2.1

   \[\leadsto \log \left(2 \cdot \frac{\frac{2}{f}}{\pi} + \color{blue}{f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)}\right) \cdot \frac{-4}{\pi} \]

6. Simplified 2.1

   \[\leadsto \log \left(2 \cdot \frac{\frac{2}{f}}{\pi} + \color{blue}{f \cdot \left(\pi \cdot 0.08333333333333333\right)}\right) \cdot \frac{-4}{\pi} \]
   Proof

   [Start] 2.1	\[ \log \left(2 \cdot \frac{\frac{2}{f}}{\pi} + f \cdot \left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right)\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-2 [=>] 2.1	\[ \log \left(2 \cdot \frac{\frac{2}{f}}{\pi} + f \cdot \left(\color{blue}{\pi \cdot 0.125} - 0.041666666666666664 \cdot \pi\right)\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-52 [=>] 2.1	\[ \log \left(2 \cdot \frac{\frac{2}{f}}{\pi} + f \cdot \color{blue}{\left(\pi \cdot \left(0.125 - 0.041666666666666664\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   metadata-eval [=>] 2.1	\[ \log \left(2 \cdot \frac{\frac{2}{f}}{\pi} + f \cdot \left(\pi \cdot \color{blue}{0.08333333333333333}\right)\right) \cdot \frac{-4}{\pi} \]

7. Applied egg-rr 2.1

   \[\leadsto \color{blue}{\log \left(\pi \cdot \left(0.08333333333333333 \cdot f\right) + \frac{\frac{4}{f}}{\pi}\right) \cdot \frac{-4}{\pi} + 0} \]

8. Simplified 2.1

   \[\leadsto \color{blue}{\log \left(f \cdot \left(0.08333333333333333 \cdot \pi\right) + \frac{\frac{4}{\pi}}{f}\right) \cdot \frac{-4}{\pi}} \]
   Proof

   [Start] 2.1	\[ \log \left(\pi \cdot \left(0.08333333333333333 \cdot f\right) + \frac{\frac{4}{f}}{\pi}\right) \cdot \frac{-4}{\pi} + 0 \]
   rational.json-simplify-4 [=>] 2.1	\[ \color{blue}{\log \left(\pi \cdot \left(0.08333333333333333 \cdot f\right) + \frac{\frac{4}{f}}{\pi}\right) \cdot \frac{-4}{\pi}} \]
   rational.json-simplify-43 [<=] 2.1	\[ \log \left(\color{blue}{f \cdot \left(\pi \cdot 0.08333333333333333\right)} + \frac{\frac{4}{f}}{\pi}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-2 [=>] 2.1	\[ \log \left(f \cdot \color{blue}{\left(0.08333333333333333 \cdot \pi\right)} + \frac{\frac{4}{f}}{\pi}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-44 [=>] 2.1	\[ \log \left(f \cdot \left(0.08333333333333333 \cdot \pi\right) + \color{blue}{\frac{\frac{4}{\pi}}{f}}\right) \cdot \frac{-4}{\pi} \]

9. Applied egg-rr 2.0

   \[\leadsto \color{blue}{\frac{\frac{\log \left(0.08333333333333333 \cdot \left(\pi \cdot f\right) + \frac{4}{\pi \cdot f}\right)}{\pi}}{-0.25}} \]

10. Final simplification 2.0

    \[\leadsto \frac{\frac{\log \left(0.08333333333333333 \cdot \left(\pi \cdot f\right) + \frac{4}{\pi \cdot f}\right)}{\pi}}{-0.25} \]

Alternatives

Alternative 1
Error	2.1
Cost	26368

\[\log \left(0.08333333333333333 \cdot \left(f \cdot \pi\right) + \frac{4}{f \cdot \pi}\right) \cdot \frac{-4}{\pi} \]

Alternative 2
Error	2.4
Cost	19648

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

Alternative 3
Error	2.4
Cost	19648

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

Alternative 4
Error	2.3
Cost	19648

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

Reproduce

herbie shell --seed 2023064 
(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)))))))))