VandenBroeck and Keller, Equation (20)

Average Error: 61.4 → 2.3

Time: 25.9s

Precision: binary64

Cost: 39232

Math

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

↓

\[-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + \pi \cdot \left({f}^{2} \cdot -0.08333333333333333\right) \]

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)) (log f)) PI))
  (* PI (* (pow f 2.0) -0.08333333333333333))))

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))) - log(f)) / ((double) M_PI))) + (((double) M_PI) * (pow(f, 2.0) * -0.08333333333333333));
}

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)) - Math.log(f)) / Math.PI)) + (Math.PI * (Math.pow(f, 2.0) * -0.08333333333333333));
}

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)) - math.log(f)) / math.pi)) + (math.pi * (math.pow(f, 2.0) * -0.08333333333333333))

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(-4.0 * Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi)) + Float64(pi * Float64((f ^ 2.0) * -0.08333333333333333)))
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)) - log(f)) / pi)) + (pi * ((f ^ 2.0) * -0.08333333333333333));
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[(-4.0 * N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] + N[(Pi * N[(N[Power[f, 2.0], $MachinePrecision] * -0.08333333333333333), $MachinePrecision]), $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. Simplified 61.4

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

   [Start] 61.4	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   rational_best_45_simplify-12 [=>] 61.4	\[ \color{blue}{\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right) \cdot -1} \]
   rational_best_45_simplify-91 [=>] 61.4	\[ \color{blue}{-1 \cdot \left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right)} \]
   rational_best_45_simplify-91 [=>] 61.4	\[ -1 \cdot \color{blue}{\left(\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \frac{1}{\frac{\pi}{4}}\right)} \]
   rational_best_45_simplify-25 [=>] 61.4	\[ \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \left(-1 \cdot \frac{1}{\frac{\pi}{4}}\right)} \]

3. Taylor expanded in f around 0 2.5

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

4. Simplified 2.5

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\color{blue}{\left(\pi \cdot 0.5\right) \cdot f + {f}^{3} \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right)}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right) \]
   Proof

   [Start] 2.5	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right) \]
   rational_best_45_simplify-91 [=>] 2.5	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\left(0.25 \cdot \pi - \color{blue}{\pi \cdot -0.25}\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right) \]
   rational_best_45_simplify-95 [=>] 2.5	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)} \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right) \]
   metadata-eval [=>] 2.5	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\left(\pi \cdot \color{blue}{0.5}\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right) \]
   rational_best_45_simplify-91 [=>] 2.5	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\left(\pi \cdot 0.5\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - \color{blue}{{\pi}^{3} \cdot -0.0026041666666666665}\right)}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right) \]
   rational_best_45_simplify-95 [=>] 2.5	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\left(\pi \cdot 0.5\right) \cdot f + {f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right) \]
   metadata-eval [=>] 2.5	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\left(\pi \cdot 0.5\right) \cdot f + {f}^{3} \cdot \left({\pi}^{3} \cdot \color{blue}{0.005208333333333333}\right)}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right) \]

5. Taylor expanded in f around 0 2.4

   \[\leadsto \color{blue}{\left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + \left(\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f\right)\right)\right)} \cdot \left(-\frac{1}{\frac{\pi}{4}}\right) \]

6. Simplified 2.4

   \[\leadsto \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(\left(-\log f\right) + \left({f}^{2} \cdot \left(0 + 0.5 \cdot \left(\pi \cdot \left(0.5 \cdot \left(\pi \cdot 0.08333333333333333\right)\right)\right)\right) + f \cdot 0\right)\right)\right)} \cdot \left(-\frac{1}{\frac{\pi}{4}}\right) \]
   Proof

   [Start] 2.4	\[ \left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + \left(\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f\right)\right)\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right) \]
   rational_best_45_simplify-80 [=>] 2.4	\[ \left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + -1 \cdot \log f\right)\right)}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right) \]
   rational_best_45_simplify-80 [=>] 2.4	\[ \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + -1 \cdot \log f\right)\right)\right)} \cdot \left(-\frac{1}{\frac{\pi}{4}}\right) \]

7. Taylor expanded in f around 0 2.3

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

8. Simplified 2.3

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

   [Start] 2.3	\[ -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + -0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right) \]
   rational_best_45_simplify-25 [=>] 2.3	\[ -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + \color{blue}{{f}^{2} \cdot \left(-0.08333333333333333 \cdot \pi\right)} \]
   rational_best_45_simplify-91 [=>] 2.3	\[ -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + {f}^{2} \cdot \color{blue}{\left(\pi \cdot -0.08333333333333333\right)} \]
   rational_best_45_simplify-25 [=>] 2.3	\[ -4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi} + \color{blue}{\pi \cdot \left({f}^{2} \cdot -0.08333333333333333\right)} \]

9. Final simplification 2.3

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

Alternatives

Alternative 1
Error	2.4
Cost	26944

\[\log \left(2 \cdot \frac{1}{\left(\pi \cdot 0.5\right) \cdot f} + \left(f \cdot \left(\pi \cdot 0.08333333333333333\right) + 0\right)\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right) \]

Alternative 2
Error	2.7
Cost	26048

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

Alternative 3
Error	2.8
Cost	19840

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

Alternative 4
Error	61.3
Cost	13120

\[-0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right) \]

Reproduce

herbie shell --seed 2023098 
(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)))))))))