VandenBroeck and Keller, Equation (20)

Percentage Accurate: 7.1% → 7.1%

Time: 40.2s

Alternatives: 6

Speedup: 1.0×

• could not determine a ground truth

  1. f = -11700944437.025612

Specification

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Initial Program: 7.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Alternative 1: 7.1% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)) (t_1 (exp t_0)) (t_2 (exp (- t_0))))
   (- (* (/ 1.0 (/ PI 4.0)) (log (/ (+ t_1 t_2) (- t_1 t_2)))))))

C

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double t_1 = exp(t_0);
	double t_2 = exp(-t_0);
	return -((1.0 / (((double) M_PI) / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
}

Java

public static double code(double f) {
	double t_0 = (Math.PI / 4.0) * f;
	double t_1 = Math.exp(t_0);
	double t_2 = Math.exp(-t_0);
	return -((1.0 / (Math.PI / 4.0)) * Math.log(((t_1 + t_2) / (t_1 - t_2))));
}

Python

def code(f):
	t_0 = (math.pi / 4.0) * f
	t_1 = math.exp(t_0)
	t_2 = math.exp(-t_0)
	return -((1.0 / (math.pi / 4.0)) * math.log(((t_1 + t_2) / (t_1 - t_2))))

Julia

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	t_1 = exp(t_0)
	t_2 = exp(Float64(-t_0))
	return Float64(-Float64(Float64(1.0 / Float64(pi / 4.0)) * log(Float64(Float64(t_1 + t_2) / Float64(t_1 - t_2)))))
end

MATLAB

function tmp = code(f)
	t_0 = (pi / 4.0) * f;
	t_1 = exp(t_0);
	t_2 = exp(-t_0);
	tmp = -((1.0 / (pi / 4.0)) * log(((t_1 + t_2) / (t_1 - t_2))));
end

Wolfram

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, Block[{t$95$1 = N[Exp[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Exp[(-t$95$0)], $MachinePrecision]}, (-N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * N[Log[N[(N[(t$95$1 + t$95$2), $MachinePrecision] / N[(t$95$1 - t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision])]]]

Derivation

1. Initial program 5.4%

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

Alternative 2: 7.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi \cdot f}{4}}\\ t_1 := {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}\\ \frac{1}{\frac{\pi}{4}} \cdot \left(-\log \left(\frac{t_0 + t_1}{t_0 - t_1}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (/ (* PI f) 4.0))) (t_1 (pow (exp (/ PI 4.0)) (- f))))
   (* (/ 1.0 (/ PI 4.0)) (- (log (/ (+ t_0 t_1) (- t_0 t_1)))))))

C

double code(double f) {
	double t_0 = exp(((((double) M_PI) * f) / 4.0));
	double t_1 = pow(exp((((double) M_PI) / 4.0)), -f);
	return (1.0 / (((double) M_PI) / 4.0)) * -log(((t_0 + t_1) / (t_0 - t_1)));
}

Java

public static double code(double f) {
	double t_0 = Math.exp(((Math.PI * f) / 4.0));
	double t_1 = Math.pow(Math.exp((Math.PI / 4.0)), -f);
	return (1.0 / (Math.PI / 4.0)) * -Math.log(((t_0 + t_1) / (t_0 - t_1)));
}

Python

def code(f):
	t_0 = math.exp(((math.pi * f) / 4.0))
	t_1 = math.pow(math.exp((math.pi / 4.0)), -f)
	return (1.0 / (math.pi / 4.0)) * -math.log(((t_0 + t_1) / (t_0 - t_1)))

Julia

function code(f)
	t_0 = exp(Float64(Float64(pi * f) / 4.0))
	t_1 = exp(Float64(pi / 4.0)) ^ Float64(-f)
	return Float64(Float64(1.0 / Float64(pi / 4.0)) * Float64(-log(Float64(Float64(t_0 + t_1) / Float64(t_0 - t_1)))))
end

MATLAB

function tmp = code(f)
	t_0 = exp(((pi * f) / 4.0));
	t_1 = exp((pi / 4.0)) ^ -f;
	tmp = (1.0 / (pi / 4.0)) * -log(((t_0 + t_1) / (t_0 - t_1)));
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(N[(Pi * f), $MachinePrecision] / 4.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Exp[N[(Pi / 4.0), $MachinePrecision]], $MachinePrecision], (-f)], $MachinePrecision]}, N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * (-N[Log[N[(N[(t$95$0 + t$95$1), $MachinePrecision] / N[(t$95$0 - t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Initial program 5.4%

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

Alternative 3: 7.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{4} \cdot \left(-f\right)}\\ t_1 := {\left(e^{f}\right)}^{\left(\frac{\pi}{4}\right)}\\ \log \left(\frac{t_0 + t_1}{t_1 - t_0}\right) \cdot \left(-\frac{4}{\pi}\right) \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (* (/ PI 4.0) (- f)))) (t_1 (pow (exp f) (/ PI 4.0))))
   (* (log (/ (+ t_0 t_1) (- t_1 t_0))) (- (/ 4.0 PI)))))

C

double code(double f) {
	double t_0 = exp(((((double) M_PI) / 4.0) * -f));
	double t_1 = pow(exp(f), (((double) M_PI) / 4.0));
	return log(((t_0 + t_1) / (t_1 - t_0))) * -(4.0 / ((double) M_PI));
}

Java

public static double code(double f) {
	double t_0 = Math.exp(((Math.PI / 4.0) * -f));
	double t_1 = Math.pow(Math.exp(f), (Math.PI / 4.0));
	return Math.log(((t_0 + t_1) / (t_1 - t_0))) * -(4.0 / Math.PI);
}

Python

def code(f):
	t_0 = math.exp(((math.pi / 4.0) * -f))
	t_1 = math.pow(math.exp(f), (math.pi / 4.0))
	return math.log(((t_0 + t_1) / (t_1 - t_0))) * -(4.0 / math.pi)

Julia

function code(f)
	t_0 = exp(Float64(Float64(pi / 4.0) * Float64(-f)))
	t_1 = exp(f) ^ Float64(pi / 4.0)
	return Float64(log(Float64(Float64(t_0 + t_1) / Float64(t_1 - t_0))) * Float64(-Float64(4.0 / pi)))
end

MATLAB

function tmp = code(f)
	t_0 = exp(((pi / 4.0) * -f));
	t_1 = exp(f) ^ (pi / 4.0);
	tmp = log(((t_0 + t_1) / (t_1 - t_0))) * -(4.0 / pi);
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Exp[f], $MachinePrecision], N[(Pi / 4.0), $MachinePrecision]], $MachinePrecision]}, N[(N[Log[N[(N[(t$95$0 + t$95$1), $MachinePrecision] / N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(4.0 / Pi), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Initial program 5.4%

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

Alternative 4: 7.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\frac{\pi}{\frac{-4}{f}}}\\ t_1 := {\left(e^{\frac{\pi}{4}}\right)}^{f}\\ \log \left(\frac{t_0 + t_1}{t_1 - t_0}\right) \cdot \frac{-4}{\pi} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (exp (/ PI (/ -4.0 f)))) (t_1 (pow (exp (/ PI 4.0)) f)))
   (* (log (/ (+ t_0 t_1) (- t_1 t_0))) (/ -4.0 PI))))

C

double code(double f) {
	double t_0 = exp((((double) M_PI) / (-4.0 / f)));
	double t_1 = pow(exp((((double) M_PI) / 4.0)), f);
	return log(((t_0 + t_1) / (t_1 - t_0))) * (-4.0 / ((double) M_PI));
}

Java

public static double code(double f) {
	double t_0 = Math.exp((Math.PI / (-4.0 / f)));
	double t_1 = Math.pow(Math.exp((Math.PI / 4.0)), f);
	return Math.log(((t_0 + t_1) / (t_1 - t_0))) * (-4.0 / Math.PI);
}

Python

def code(f):
	t_0 = math.exp((math.pi / (-4.0 / f)))
	t_1 = math.pow(math.exp((math.pi / 4.0)), f)
	return math.log(((t_0 + t_1) / (t_1 - t_0))) * (-4.0 / math.pi)

Julia

function code(f)
	t_0 = exp(Float64(pi / Float64(-4.0 / f)))
	t_1 = exp(Float64(pi / 4.0)) ^ f
	return Float64(log(Float64(Float64(t_0 + t_1) / Float64(t_1 - t_0))) * Float64(-4.0 / pi))
end

MATLAB

function tmp = code(f)
	t_0 = exp((pi / (-4.0 / f)));
	t_1 = exp((pi / 4.0)) ^ f;
	tmp = log(((t_0 + t_1) / (t_1 - t_0))) * (-4.0 / pi);
end

Wolfram

code[f_] := Block[{t$95$0 = N[Exp[N[(Pi / N[(-4.0 / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Exp[N[(Pi / 4.0), $MachinePrecision]], $MachinePrecision], f], $MachinePrecision]}, N[(N[Log[N[(N[(t$95$0 + t$95$1), $MachinePrecision] / N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 5.4%

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

Alternative 5: 7.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{f}\right)}^{\left(\frac{\pi}{-4}\right)}\\ t_1 := {\left(e^{\frac{\pi}{4}}\right)}^{f}\\ \log \left(\frac{t_0 + t_1}{t_1 - t_0}\right) \cdot \frac{-4}{\pi} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (pow (exp f) (/ PI -4.0))) (t_1 (pow (exp (/ PI 4.0)) f)))
   (* (log (/ (+ t_0 t_1) (- t_1 t_0))) (/ -4.0 PI))))

C

double code(double f) {
	double t_0 = pow(exp(f), (((double) M_PI) / -4.0));
	double t_1 = pow(exp((((double) M_PI) / 4.0)), f);
	return log(((t_0 + t_1) / (t_1 - t_0))) * (-4.0 / ((double) M_PI));
}

Java

public static double code(double f) {
	double t_0 = Math.pow(Math.exp(f), (Math.PI / -4.0));
	double t_1 = Math.pow(Math.exp((Math.PI / 4.0)), f);
	return Math.log(((t_0 + t_1) / (t_1 - t_0))) * (-4.0 / Math.PI);
}

Python

def code(f):
	t_0 = math.pow(math.exp(f), (math.pi / -4.0))
	t_1 = math.pow(math.exp((math.pi / 4.0)), f)
	return math.log(((t_0 + t_1) / (t_1 - t_0))) * (-4.0 / math.pi)

Julia

function code(f)
	t_0 = exp(f) ^ Float64(pi / -4.0)
	t_1 = exp(Float64(pi / 4.0)) ^ f
	return Float64(log(Float64(Float64(t_0 + t_1) / Float64(t_1 - t_0))) * Float64(-4.0 / pi))
end

MATLAB

function tmp = code(f)
	t_0 = exp(f) ^ (pi / -4.0);
	t_1 = exp((pi / 4.0)) ^ f;
	tmp = log(((t_0 + t_1) / (t_1 - t_0))) * (-4.0 / pi);
end

Wolfram

code[f_] := Block[{t$95$0 = N[Power[N[Exp[f], $MachinePrecision], N[(Pi / -4.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Exp[N[(Pi / 4.0), $MachinePrecision]], $MachinePrecision], f], $MachinePrecision]}, N[(N[Log[N[(N[(t$95$0 + t$95$1), $MachinePrecision] / N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 5.4%

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

Alternative 6: 6.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{f}\right)}^{\left(-0.25 \cdot \pi\right)}\\ t_1 := {\left(e^{f}\right)}^{\left(0.25 \cdot \pi\right)}\\ \log \left(\frac{t_0 + t_1}{t_1 - t_0}\right) \cdot \frac{-4}{\pi} \end{array} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (let* ((t_0 (pow (exp f) (* -0.25 PI))) (t_1 (pow (exp f) (* 0.25 PI))))
   (* (log (/ (+ t_0 t_1) (- t_1 t_0))) (/ -4.0 PI))))

C

double code(double f) {
	double t_0 = pow(exp(f), (-0.25 * ((double) M_PI)));
	double t_1 = pow(exp(f), (0.25 * ((double) M_PI)));
	return log(((t_0 + t_1) / (t_1 - t_0))) * (-4.0 / ((double) M_PI));
}

Java

public static double code(double f) {
	double t_0 = Math.pow(Math.exp(f), (-0.25 * Math.PI));
	double t_1 = Math.pow(Math.exp(f), (0.25 * Math.PI));
	return Math.log(((t_0 + t_1) / (t_1 - t_0))) * (-4.0 / Math.PI);
}

Python

def code(f):
	t_0 = math.pow(math.exp(f), (-0.25 * math.pi))
	t_1 = math.pow(math.exp(f), (0.25 * math.pi))
	return math.log(((t_0 + t_1) / (t_1 - t_0))) * (-4.0 / math.pi)

Julia

function code(f)
	t_0 = exp(f) ^ Float64(-0.25 * pi)
	t_1 = exp(f) ^ Float64(0.25 * pi)
	return Float64(log(Float64(Float64(t_0 + t_1) / Float64(t_1 - t_0))) * Float64(-4.0 / pi))
end

MATLAB

function tmp = code(f)
	t_0 = exp(f) ^ (-0.25 * pi);
	t_1 = exp(f) ^ (0.25 * pi);
	tmp = log(((t_0 + t_1) / (t_1 - t_0))) * (-4.0 / pi);
end

Wolfram

code[f_] := Block[{t$95$0 = N[Power[N[Exp[f], $MachinePrecision], N[(-0.25 * Pi), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Exp[f], $MachinePrecision], N[(0.25 * Pi), $MachinePrecision]], $MachinePrecision]}, N[(N[Log[N[(N[(t$95$0 + t$95$1), $MachinePrecision] / N[(t$95$1 - t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 5.4%

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

Reproduce

herbie shell --seed 2023276 
(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)))))))))