VandenBroeck and Keller, Equation (20)

Average Error: 61.5 → 2.1

Time: 20.5s

Precision: binary64

Cost: 104832

Math

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

↓

\[\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{{f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right) + \left({f}^{3} \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right) + \left(\pi \cdot \left(f \cdot 0.5\right) + {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

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
   (/
    (+ (exp (* (/ PI 4.0) f)) (exp (* f (/ PI -4.0))))
    (+
     (* (pow f 5.0) (* (pow PI 5.0) 1.6276041666666666e-5))
     (+
      (* (pow f 3.0) (* (pow PI 3.0) 0.005208333333333333))
      (+
       (* PI (* f 0.5))
       (* (pow f 7.0) (* (pow PI 7.0) 2.422030009920635e-8)))))))
  (/ -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(((exp(((((double) M_PI) / 4.0) * f)) + exp((f * (((double) M_PI) / -4.0)))) / ((pow(f, 5.0) * (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5)) + ((pow(f, 3.0) * (pow(((double) M_PI), 3.0) * 0.005208333333333333)) + ((((double) M_PI) * (f * 0.5)) + (pow(f, 7.0) * (pow(((double) M_PI), 7.0) * 2.422030009920635e-8))))))) * (-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.exp(((Math.PI / 4.0) * f)) + Math.exp((f * (Math.PI / -4.0)))) / ((Math.pow(f, 5.0) * (Math.pow(Math.PI, 5.0) * 1.6276041666666666e-5)) + ((Math.pow(f, 3.0) * (Math.pow(Math.PI, 3.0) * 0.005208333333333333)) + ((Math.PI * (f * 0.5)) + (Math.pow(f, 7.0) * (Math.pow(Math.PI, 7.0) * 2.422030009920635e-8))))))) * (-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.exp(((math.pi / 4.0) * f)) + math.exp((f * (math.pi / -4.0)))) / ((math.pow(f, 5.0) * (math.pow(math.pi, 5.0) * 1.6276041666666666e-5)) + ((math.pow(f, 3.0) * (math.pow(math.pi, 3.0) * 0.005208333333333333)) + ((math.pi * (f * 0.5)) + (math.pow(f, 7.0) * (math.pow(math.pi, 7.0) * 2.422030009920635e-8))))))) * (-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(log(Float64(Float64(exp(Float64(Float64(pi / 4.0) * f)) + exp(Float64(f * Float64(pi / -4.0)))) / Float64(Float64((f ^ 5.0) * Float64((pi ^ 5.0) * 1.6276041666666666e-5)) + Float64(Float64((f ^ 3.0) * Float64((pi ^ 3.0) * 0.005208333333333333)) + Float64(Float64(pi * Float64(f * 0.5)) + Float64((f ^ 7.0) * Float64((pi ^ 7.0) * 2.422030009920635e-8))))))) * Float64(-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(((exp(((pi / 4.0) * f)) + exp((f * (pi / -4.0)))) / (((f ^ 5.0) * ((pi ^ 5.0) * 1.6276041666666666e-5)) + (((f ^ 3.0) * ((pi ^ 3.0) * 0.005208333333333333)) + ((pi * (f * 0.5)) + ((f ^ 7.0) * ((pi ^ 7.0) * 2.422030009920635e-8))))))) * (-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[Log[N[(N[(N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(f * N[(Pi / -4.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision]), $MachinePrecision] + N[(N[(Pi * N[(f * 0.5), $MachinePrecision]), $MachinePrecision] + N[(N[Power[f, 7.0], $MachinePrecision] * N[(N[Power[Pi, 7.0], $MachinePrecision] * 2.422030009920635e-8), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $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

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

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{\color{blue}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

4. Simplified 2.1

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{\color{blue}{{f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right) + \left({f}^{3} \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right) + \left(\pi \cdot \left(f \cdot 0.5\right) + {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)}}\right) \cdot \frac{-4}{\pi} \]
   Proof

   [Start] 2.1	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-2 [=>] 2.1	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{{f}^{5} \cdot \left(\color{blue}{{\pi}^{5} \cdot 8.138020833333333 \cdot 10^{-6}} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-52 [=>] 2.1	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{{f}^{5} \cdot \color{blue}{\left({\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)\right)} + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   metadata-eval [=>] 2.1	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{{f}^{5} \cdot \left({\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-4 [<=] 2.1	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{{f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \color{blue}{\left(\left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right) + 0\right)}\right)}\right) \cdot \frac{-4}{\pi} \]
   rational.json-simplify-41 [<=] 2.1	\[ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{{f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right) + \color{blue}{\left(0 + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)\right)}}\right) \cdot \frac{-4}{\pi} \]

5. Final simplification 2.1

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \frac{\pi}{-4}}}{{f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right) + \left({f}^{3} \cdot \left({\pi}^{3} \cdot 0.005208333333333333\right) + \left(\pi \cdot \left(f \cdot 0.5\right) + {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \]

Alternatives

Alternative 1
Error	2.2
Cost	59072

\[\frac{\frac{\log \left(\frac{e^{\pi \cdot \left(f \cdot 0.25\right)} + e^{\pi \cdot \left(f \cdot -0.25\right)}}{\pi \cdot \left(f \cdot 0.5\right) + 0.005208333333333333 \cdot {\left(\pi \cdot f\right)}^{3}}\right)}{\pi}}{-0.25} \]

Alternative 2
Error	2.2
Cost	58944

\[-4 \cdot \frac{\left(-\log f\right) + \log \left(\frac{4}{\pi}\right)}{\pi} + \left({f}^{2} \cdot \left(\pi \cdot -0.08333333333333333\right) + {\pi}^{3} \cdot \left({f}^{4} \cdot 0.0008680555555555555\right)\right) \]

Alternative 3
Error	2.3
Cost	26496

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

Alternative 4
Error	2.3
Cost	26496

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

Alternative 5
Error	2.5
Cost	19968

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

Alternative 6
Error	2.6
Cost	19648

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

Reproduce

herbie shell --seed 2023073 
(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)))))))))