VandenBroeck and Keller, Equation (20)

Average Error: 61.5 → 2.3

Time: 21.1s

Precision: binary64

Cost: 39232

Math

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

↓

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

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 f) (log (/ 4.0 PI))) PI))
  (* -0.08333333333333333 (* (pow f 2.0) 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 (4.0 * ((log(f) - log((4.0 / ((double) M_PI)))) / ((double) M_PI))) + (-0.08333333333333333 * (pow(f, 2.0) * ((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 (4.0 * ((Math.log(f) - Math.log((4.0 / Math.PI))) / Math.PI)) + (-0.08333333333333333 * (Math.pow(f, 2.0) * 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 (4.0 * ((math.log(f) - math.log((4.0 / math.pi))) / math.pi)) + (-0.08333333333333333 * (math.pow(f, 2.0) * 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(4.0 * Float64(Float64(log(f) - log(Float64(4.0 / pi))) / pi)) + Float64(-0.08333333333333333 * Float64((f ^ 2.0) * 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 = (4.0 * ((log(f) - log((4.0 / pi))) / pi)) + (-0.08333333333333333 * ((f ^ 2.0) * 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[(4.0 * N[(N[(N[Log[f], $MachinePrecision] - N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision] + N[(-0.08333333333333333 * N[(N[Power[f, 2.0], $MachinePrecision] * Pi), $MachinePrecision]), $MachinePrecision]), $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

   \[\leadsto \color{blue}{\frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{e^{\frac{\pi}{4} \cdot f} - e^{\frac{\pi}{4} \cdot \left(-f\right)}}\right)\right)} \]
   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_best_oopsla_all_46_json_45_simplify-94 [=>] 61.5	\[ \color{blue}{\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right) \cdot -1} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 61.5	\[ \color{blue}{-1 \cdot \left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-7 [=>] 61.5	\[ \color{blue}{\frac{1}{\frac{\pi}{4}} \cdot \left(-1 \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 61.5	\[ \frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot -1\right)} \]
   rational_best_oopsla_all_46_json_45_simplify-92 [=>] 61.5	\[ \frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right)\right)} \]

3. Taylor expanded in f around 0 2.3

   \[\leadsto \frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}}\right)\right) \]

4. Simplified 2.3

   \[\leadsto \frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\color{blue}{\left(\pi \cdot 0.5\right) \cdot f + \left({f}^{3} \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right) + {f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)}}\right)\right) \]
   Proof

   [Start] 2.3	\[ \frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-82 [=>] 2.3	\[ \frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\color{blue}{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}}\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 2.3	\[ \frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\left(0.25 \cdot \pi - \color{blue}{\pi \cdot -0.25}\right) \cdot f + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-102 [=>] 2.3	\[ \frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\color{blue}{\left(\pi \cdot \left(0.25 - -0.25\right)\right)} \cdot f + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)\right) \]
   metadata-eval [=>] 2.3	\[ \frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\left(\pi \cdot \color{blue}{0.5}\right) \cdot f + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)}\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 2.3	\[ \frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\left(\pi \cdot 0.5\right) \cdot f + \color{blue}{\left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}}\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 2.3	\[ \frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\left(\pi \cdot 0.5\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - \color{blue}{{\pi}^{3} \cdot -0.0026041666666666665}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-102 [=>] 2.3	\[ \frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\left(\pi \cdot 0.5\right) \cdot f + \left({f}^{3} \cdot \color{blue}{\left({\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)\right)} + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right)\right) \]
   metadata-eval [=>] 2.3	\[ \frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\left(\pi \cdot 0.5\right) \cdot f + \left({f}^{3} \cdot \left({\pi}^{3} \cdot \color{blue}{0.005208333333333333}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right)\right) \]

5. Taylor expanded in f around 0 2.4

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

6. Simplified 2.4

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

   [Start] 2.4	\[ \frac{1}{\frac{\pi}{4}} \cdot \left(-\left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + \left(\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f\right)\right)\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-82 [=>] 2.4	\[ \frac{1}{\frac{\pi}{4}} \cdot \left(-\left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + -1 \cdot \log f\right)\right)}\right)\right) \]
   rational_best_oopsla_all_46_json_45_simplify-82 [=>] 2.4	\[ \frac{1}{\frac{\pi}{4}} \cdot \left(-\color{blue}{\left(\log \left(\frac{4}{\pi}\right) + \left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + -1 \cdot \log f\right)\right)\right)}\right) \]
   rational_best_oopsla_all_46_json_45_simplify-35 [=>] 2.4	\[ \frac{1}{\frac{\pi}{4}} \cdot \left(-\left(\log \left(\frac{4}{\pi}\right) + \left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \color{blue}{\left(-1 \cdot \log f + 0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right)\right)}\right)\right)\right) \]

7. Taylor expanded in f around 0 2.3

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

8. Final simplification 2.3

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

Alternatives

Alternative 1
Error	2.7
Cost	26176

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

Alternative 2
Error	2.6
Cost	26048

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

Alternative 3
Error	2.8
Cost	19840

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

Alternative 4
Error	61.3
Cost	13120

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

Alternative 5
Error	61.3
Cost	13120

\[\pi \cdot \left(-0.125 \cdot {f}^{2}\right) \]

Reproduce

herbie shell --seed 2023090 
(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)))))))))