VandenBroeck and Keller, Equation (20)

Average Error: 95.98% → 3.36%

Time: 20.5s

Precision: binary64

Cost: 150592

Math

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

↓

\[\begin{array}{l} t_0 := 8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\\ t_1 := {\pi}^{3} \cdot -0.0026041666666666665\\ t_2 := {\pi}^{7} \cdot -1.2110150049603175 \cdot 10^{-8}\\ \log \left(\frac{e^{\frac{\pi \cdot f}{4}} + e^{\left(\pi \cdot f\right) \cdot -0.25}}{{f}^{5} \cdot \left(t_0 + t_0\right) - \left(f \cdot \left(\pi \cdot -0.25 + \pi \cdot -0.25\right) + \left({f}^{7} \cdot \left(t_2 + t_2\right) + {f}^{3} \cdot \left(t_1 + t_1\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \end{array} \]

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
 (let* ((t_0 (* 8.138020833333333e-6 (pow PI 5.0)))
        (t_1 (* (pow PI 3.0) -0.0026041666666666665))
        (t_2 (* (pow PI 7.0) -1.2110150049603175e-8)))
   (*
    (log
     (/
      (+ (exp (/ (* PI f) 4.0)) (exp (* (* PI f) -0.25)))
      (-
       (* (pow f 5.0) (+ t_0 t_0))
       (+
        (* f (+ (* PI -0.25) (* PI -0.25)))
        (+ (* (pow f 7.0) (+ t_2 t_2)) (* (pow f 3.0) (+ t_1 t_1)))))))
    (/ -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) {
	double t_0 = 8.138020833333333e-6 * pow(((double) M_PI), 5.0);
	double t_1 = pow(((double) M_PI), 3.0) * -0.0026041666666666665;
	double t_2 = pow(((double) M_PI), 7.0) * -1.2110150049603175e-8;
	return log(((exp(((((double) M_PI) * f) / 4.0)) + exp(((((double) M_PI) * f) * -0.25))) / ((pow(f, 5.0) * (t_0 + t_0)) - ((f * ((((double) M_PI) * -0.25) + (((double) M_PI) * -0.25))) + ((pow(f, 7.0) * (t_2 + t_2)) + (pow(f, 3.0) * (t_1 + t_1))))))) * (-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) {
	double t_0 = 8.138020833333333e-6 * Math.pow(Math.PI, 5.0);
	double t_1 = Math.pow(Math.PI, 3.0) * -0.0026041666666666665;
	double t_2 = Math.pow(Math.PI, 7.0) * -1.2110150049603175e-8;
	return Math.log(((Math.exp(((Math.PI * f) / 4.0)) + Math.exp(((Math.PI * f) * -0.25))) / ((Math.pow(f, 5.0) * (t_0 + t_0)) - ((f * ((Math.PI * -0.25) + (Math.PI * -0.25))) + ((Math.pow(f, 7.0) * (t_2 + t_2)) + (Math.pow(f, 3.0) * (t_1 + t_1))))))) * (-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):
	t_0 = 8.138020833333333e-6 * math.pow(math.pi, 5.0)
	t_1 = math.pow(math.pi, 3.0) * -0.0026041666666666665
	t_2 = math.pow(math.pi, 7.0) * -1.2110150049603175e-8
	return math.log(((math.exp(((math.pi * f) / 4.0)) + math.exp(((math.pi * f) * -0.25))) / ((math.pow(f, 5.0) * (t_0 + t_0)) - ((f * ((math.pi * -0.25) + (math.pi * -0.25))) + ((math.pow(f, 7.0) * (t_2 + t_2)) + (math.pow(f, 3.0) * (t_1 + t_1))))))) * (-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)
	t_0 = Float64(8.138020833333333e-6 * (pi ^ 5.0))
	t_1 = Float64((pi ^ 3.0) * -0.0026041666666666665)
	t_2 = Float64((pi ^ 7.0) * -1.2110150049603175e-8)
	return Float64(log(Float64(Float64(exp(Float64(Float64(pi * f) / 4.0)) + exp(Float64(Float64(pi * f) * -0.25))) / Float64(Float64((f ^ 5.0) * Float64(t_0 + t_0)) - Float64(Float64(f * Float64(Float64(pi * -0.25) + Float64(pi * -0.25))) + Float64(Float64((f ^ 7.0) * Float64(t_2 + t_2)) + Float64((f ^ 3.0) * Float64(t_1 + t_1))))))) * 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)
	t_0 = 8.138020833333333e-6 * (pi ^ 5.0);
	t_1 = (pi ^ 3.0) * -0.0026041666666666665;
	t_2 = (pi ^ 7.0) * -1.2110150049603175e-8;
	tmp = log(((exp(((pi * f) / 4.0)) + exp(((pi * f) * -0.25))) / (((f ^ 5.0) * (t_0 + t_0)) - ((f * ((pi * -0.25) + (pi * -0.25))) + (((f ^ 7.0) * (t_2 + t_2)) + ((f ^ 3.0) * (t_1 + t_1))))))) * (-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_] := Block[{t$95$0 = N[(8.138020833333333e-6 * N[Power[Pi, 5.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[Pi, 3.0], $MachinePrecision] * -0.0026041666666666665), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[Pi, 7.0], $MachinePrecision] * -1.2110150049603175e-8), $MachinePrecision]}, N[(N[Log[N[(N[(N[Exp[N[(N[(Pi * f), $MachinePrecision] / 4.0), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(Pi * f), $MachinePrecision] * -0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[Power[f, 5.0], $MachinePrecision] * N[(t$95$0 + t$95$0), $MachinePrecision]), $MachinePrecision] - N[(N[(f * N[(N[(Pi * -0.25), $MachinePrecision] + N[(Pi * -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[f, 7.0], $MachinePrecision] * N[(t$95$2 + t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[Power[f, 3.0], $MachinePrecision] * N[(t$95$1 + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 95.98

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

2. Simplified 95.98

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

   [Start] 95.98	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
   *-commutative [=>] 95.98	\[ -\color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \frac{1}{\frac{\pi}{4}}} \]
   distribute-rgt-neg-in [=>] 95.98	\[ \color{blue}{\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \cdot \left(-\frac{1}{\frac{\pi}{4}}\right)} \]

3. Taylor expanded in f around 0 3.36

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

4. Final simplification 3.36

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

Alternatives

Alternative 1
Error	3.65%
Cost	138304

\[\begin{array}{l} t_0 := \pi \cdot 0.25 + \pi \cdot 0.25\\ t_1 := \frac{\pi}{t_0}\\ t_2 := {\pi}^{3} \cdot -0.0026041666666666665\\ \frac{-4}{\pi} \cdot \log \left(-0.25 \cdot t_1 + \left(2 \cdot \frac{1}{f \cdot t_0} + \left(0.25 \cdot t_1 + f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{t_0} + 2 \cdot \frac{t_2 + t_2}{{t_0}^{2}}\right)\right)\right)\right) \end{array} \]

Alternative 2
Error	3.65%
Cost	78208

\[\frac{-4}{\pi} \cdot \left(\log \left(e^{\pi \cdot \left(f \cdot 0.25\right)} + e^{\pi \cdot \left(f \cdot -0.25\right)}\right) - \log \left(\mathsf{fma}\left(\pi \cdot 0.5, f, {\pi}^{3} \cdot \left({f}^{3} \cdot 0.005208333333333333\right)\right)\right)\right) \]

Alternative 3
Error	3.68%
Cost	71808

\[\frac{-4}{\pi} \cdot \log \left(\frac{e^{\pi \cdot \left(f \cdot 0.25\right)} + e^{\pi \cdot \left(f \cdot -0.25\right)}}{\mathsf{fma}\left(\pi \cdot 0.5, f, {\pi}^{3} \cdot \left({f}^{3} \cdot 0.005208333333333333\right)\right)}\right) \]

Alternative 4
Error	4.06%
Cost	32704

\[\frac{-4}{\pi} \cdot \log \left(\frac{2}{{\left({\left(\pi \cdot \left(f \cdot 0.5\right)\right)}^{0.3333333333333333}\right)}^{3}}\right) \]

Alternative 5
Error	69.42%
Cost	19648

\[\frac{-4}{\pi} \cdot \log \left(\frac{8}{\pi \cdot f}\right) \]

Alternative 6
Error	4.4%
Cost	19648

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

Alternative 7
Error	4.4%
Cost	19648

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

Alternative 8
Error	4.14%
Cost	19648

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

Alternative 9
Error	100%
Cost	13056

\[\frac{-4}{\pi} \cdot \mathsf{log1p}\left(-2\right) \]

Reproduce

herbie shell --seed 2023089 
(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)))))))))