VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 96.6%

Time: 42.2s

Alternatives: 11

Speedup: 4.9×

• could not determine a ground truth

  1. f = 2.00000000000000007e154

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: 6.9% 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: 96.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \left(-\frac{\pi}{4}\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (+ (exp (* (/ PI 4.0) f)) (exp (* f (- (/ PI 4.0)))))
    (fma
     f
     (* PI 0.5)
     (fma
      (pow f 5.0)
      (* (pow PI 5.0) 1.6276041666666666e-5)
      (fma
       (pow f 3.0)
       (* (pow PI 3.0) 0.005208333333333333)
       (* (pow f 7.0) (* (pow PI 7.0) 2.422030009920635e-8)))))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(((exp(((((double) M_PI) / 4.0) * f)) + exp((f * -(((double) M_PI) / 4.0)))) / fma(f, (((double) M_PI) * 0.5), fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), (pow(f, 7.0) * (pow(((double) M_PI), 7.0) * 2.422030009920635e-8))))))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(Float64(Float64(exp(Float64(Float64(pi / 4.0) * f)) + exp(Float64(f * Float64(-Float64(pi / 4.0))))) / fma(f, Float64(pi * 0.5), fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), Float64((f ^ 7.0) * Float64((pi ^ 7.0) * 2.422030009920635e-8))))))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

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[(f * N[(Pi * 0.5), $MachinePrecision] + N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $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[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in f around 0 97.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]
4. Step-by-step derivation

   1. fma-def 97.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, 0.25 \cdot \pi - -0.25 \cdot \pi, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]

   2. distribute-rgt-out-- 97.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   3. metadata-eval 97.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   4. associate-+r+ 97.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)}\right)}\right) \]

   5. +-commutative 97.5%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right)\right)} + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right) \]

5. Simplified 97.5%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)}}\right) \]

6. Final simplification 97.5%

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \left(-\frac{\pi}{4}\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
7. Add Preprocessing

Alternative 2: 96.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \left(-\frac{\pi}{4}\right)}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left(f, \pi \cdot 0.5, {f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    (+ (exp (* (/ PI 4.0) f)) (exp (* f (- (/ PI 4.0)))))
    (fma
     (pow f 3.0)
     (* (pow PI 3.0) 0.005208333333333333)
     (fma
      f
      (* PI 0.5)
      (* (pow f 5.0) (* (pow PI 5.0) 1.6276041666666666e-5))))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(((exp(((((double) M_PI) / 4.0) * f)) + exp((f * -(((double) M_PI) / 4.0)))) / fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), fma(f, (((double) M_PI) * 0.5), (pow(f, 5.0) * (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5)))))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(Float64(Float64(exp(Float64(Float64(pi / 4.0) * f)) + exp(Float64(f * Float64(-Float64(pi / 4.0))))) / fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), fma(f, Float64(pi * 0.5), Float64((f ^ 5.0) * Float64((pi ^ 5.0) * 1.6276041666666666e-5)))))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

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[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(f * N[(Pi * 0.5), $MachinePrecision] + N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in f around 0 97.4%

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

   1. associate-+r+ 97.4%

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

   2. +-commutative 97.4%

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

   3. associate-+l+ 97.4%

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

   4. fma-def 97.4%

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

   5. distribute-rgt-out-- 97.4%

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

   6. metadata-eval 97.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right) \]

5. Simplified 97.4%

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

6. Final simplification 97.4%

   \[\leadsto \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{f \cdot \left(-\frac{\pi}{4}\right)}}{\mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left(f, \pi \cdot 0.5, {f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
7. Add Preprocessing

Alternative 3: 96.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right), -2, 0\right)\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (fma
  (/ (- (log (/ 4.0 PI)) (log f)) PI)
  -4.0
  (fma
   (*
    (/ (pow f 2.0) PI)
    (fma
     PI
     (*
      0.5
      (fma
       (/ 0.005208333333333333 (* 0.5 (/ 0.5 PI)))
       -2.0
       (* 0.0625 (* PI 2.0))))
     0.0))
   -2.0
   0.0)))

C

double code(double f) {
	return fma(((log((4.0 / ((double) M_PI))) - log(f)) / ((double) M_PI)), -4.0, fma(((pow(f, 2.0) / ((double) M_PI)) * fma(((double) M_PI), (0.5 * fma((0.005208333333333333 / (0.5 * (0.5 / ((double) M_PI)))), -2.0, (0.0625 * (((double) M_PI) * 2.0)))), 0.0)), -2.0, 0.0));
}

Julia

function code(f)
	return fma(Float64(Float64(log(Float64(4.0 / pi)) - log(f)) / pi), -4.0, fma(Float64(Float64((f ^ 2.0) / pi) * fma(pi, Float64(0.5 * fma(Float64(0.005208333333333333 / Float64(0.5 * Float64(0.5 / pi))), -2.0, Float64(0.0625 * Float64(pi * 2.0)))), 0.0)), -2.0, 0.0))
end

Wolfram

code[f_] := N[(N[(N[(N[Log[N[(4.0 / Pi), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision] * -4.0 + N[(N[(N[(N[Power[f, 2.0], $MachinePrecision] / Pi), $MachinePrecision] * N[(Pi * N[(0.5 * N[(N[(0.005208333333333333 / N[(0.5 * N[(0.5 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(0.0625 * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision] * -2.0 + 0.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

3. Simplified 6.9%

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

5. Taylor expanded in f around 0 97.4%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi} + \left(-2 \cdot \frac{f \cdot \left(\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}{\pi} + -2 \cdot \frac{{f}^{2} \cdot \left(-0.25 \cdot \left({\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}^{2} \cdot {\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}\right) + \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right) \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)\right)}{\pi}\right)} \]

7. Simplified 97.4%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(2 \cdot \pi\right)\right), 0\right), -2, 0\right)\right)} \]

8. Final simplification 97.4%

   \[\leadsto \mathsf{fma}\left(\frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}, -4, \mathsf{fma}\left(\frac{{f}^{2}}{\pi} \cdot \mathsf{fma}\left(\pi, 0.5 \cdot \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right), 0\right), -2, 0\right)\right) \]
9. Add Preprocessing

Alternative 4: 96.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right), \frac{\frac{4}{\pi}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (fma
    f
    (fma
     (/ 0.005208333333333333 (* 0.5 (/ 0.5 PI)))
     -2.0
     (* 0.0625 (* PI 2.0)))
    (/ (/ 4.0 PI) f)))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(fma(f, fma((0.005208333333333333 / (0.5 * (0.5 / ((double) M_PI)))), -2.0, (0.0625 * (((double) M_PI) * 2.0))), ((4.0 / ((double) M_PI)) / f))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(fma(f, fma(Float64(0.005208333333333333 / Float64(0.5 * Float64(0.5 / pi))), -2.0, Float64(0.0625 * Float64(pi * 2.0))), Float64(Float64(4.0 / pi) / f))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(N[(0.005208333333333333 / N[(0.5 * N[(0.5 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0 + N[(0.0625 * N[(Pi * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in f around 0 97.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + \left(f \cdot \left(0.0625 \cdot \frac{{\pi}^{2}}{0.25 \cdot \pi - -0.25 \cdot \pi} - 2 \cdot \frac{0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}}{{\left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}^{2}}\right) + 2 \cdot \frac{1}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}\right)\right)\right)} \]

5. Simplified 97.3%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(2 \cdot \pi\right)\right), \frac{\frac{4}{\pi}}{f}\right)\right)} \]

6. Final simplification 97.3%

   \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{0.005208333333333333}{0.5 \cdot \frac{0.5}{\pi}}, -2, 0.0625 \cdot \left(\pi \cdot 2\right)\right), \frac{\frac{4}{\pi}}{f}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
7. Add Preprocessing

Alternative 5: 96.0% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi} - 0.125 \cdot \left(\pi \cdot {f}^{2}\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (- (* 4.0 (/ (- (log f) (log (/ 4.0 PI))) PI)) (* 0.125 (* PI (pow f 2.0)))))

C

double code(double f) {
	return (4.0 * ((log(f) - log((4.0 / ((double) M_PI)))) / ((double) M_PI))) - (0.125 * (((double) M_PI) * pow(f, 2.0)));
}

Java

public static double code(double f) {
	return (4.0 * ((Math.log(f) - Math.log((4.0 / Math.PI))) / Math.PI)) - (0.125 * (Math.PI * Math.pow(f, 2.0)));
}

Python

def code(f):
	return (4.0 * ((math.log(f) - math.log((4.0 / math.pi))) / math.pi)) - (0.125 * (math.pi * math.pow(f, 2.0)))

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(Float64(log(f) - log(Float64(4.0 / pi))) / pi)) - Float64(0.125 * Float64(pi * (f ^ 2.0))))
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * ((log(f) - log((4.0 / pi))) / pi)) - (0.125 * (pi * (f ^ 2.0)));
end

Wolfram

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[(0.125 * N[(Pi * N[Power[f, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in f around 0 96.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}}\right) \]
4. Step-by-step derivation

   1. distribute-rgt-out-- 96.6%

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

   2. metadata-eval 96.6%

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

5. Simplified 96.6%

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

6. Taylor expanded in f around 0 96.8%

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

7. Final simplification 96.8%

   \[\leadsto 4 \cdot \frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi} - 0.125 \cdot \left(\pi \cdot {f}^{2}\right) \]
8. Add Preprocessing

Alternative 6: 96.0% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{-\log \left(\mathsf{fma}\left(0.125, \pi \cdot f, \frac{\frac{4}{\pi}}{f}\right)\right)}{\pi \cdot 0.25} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (- (log (fma 0.125 (* PI f) (/ (/ 4.0 PI) f)))) (* PI 0.25)))

C

double code(double f) {
	return -log(fma(0.125, (((double) M_PI) * f), ((4.0 / ((double) M_PI)) / f))) / (((double) M_PI) * 0.25);
}

Julia

function code(f)
	return Float64(Float64(-log(fma(0.125, Float64(pi * f), Float64(Float64(4.0 / pi) / f)))) / Float64(pi * 0.25))
end

Wolfram

code[f_] := N[((-N[Log[N[(0.125 * N[(Pi * f), $MachinePrecision] + N[(N[(4.0 / Pi), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(Pi * 0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in f around 0 96.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}}\right) \]
4. Step-by-step derivation

   1. distribute-rgt-out-- 96.6%

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

   2. metadata-eval 96.6%

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

5. Simplified 96.6%

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

6. Taylor expanded in f around 0 96.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(0.125 \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)} \]
7. Step-by-step derivation

   1. rem-cbrt-cube 96.3%

      \[\leadsto -\frac{1}{\frac{\color{blue}{\sqrt[3]{{\pi}^{3}}}}{4}} \cdot \log \left(0.125 \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right) \]

8. Applied egg-rr 96.3%

   \[\leadsto -\frac{1}{\frac{\color{blue}{\sqrt[3]{{\pi}^{3}}}}{4}} \cdot \log \left(0.125 \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right) \]
9. Step-by-step derivation

   1. associate-*l/ 96.4%

      \[\leadsto -\color{blue}{\frac{1 \cdot \log \left(0.125 \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)}{\frac{\sqrt[3]{{\pi}^{3}}}{4}}} \]

   2. *-un-lft-identity 96.4%

      \[\leadsto -\frac{\color{blue}{\log \left(0.125 \cdot \left(f \cdot \pi\right) + 4 \cdot \frac{1}{f \cdot \pi}\right)}}{\frac{\sqrt[3]{{\pi}^{3}}}{4}} \]

   3. fma-def 96.4%

      \[\leadsto -\frac{\log \color{blue}{\left(\mathsf{fma}\left(0.125, f \cdot \pi, 4 \cdot \frac{1}{f \cdot \pi}\right)\right)}}{\frac{\sqrt[3]{{\pi}^{3}}}{4}} \]

   4. *-commutative 96.4%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(0.125, \color{blue}{\pi \cdot f}, 4 \cdot \frac{1}{f \cdot \pi}\right)\right)}{\frac{\sqrt[3]{{\pi}^{3}}}{4}} \]

   5. *-commutative 96.4%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(0.125, \pi \cdot f, 4 \cdot \frac{1}{\color{blue}{\pi \cdot f}}\right)\right)}{\frac{\sqrt[3]{{\pi}^{3}}}{4}} \]

   6. div-inv 96.4%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(0.125, \pi \cdot f, \color{blue}{\frac{4}{\pi \cdot f}}\right)\right)}{\frac{\sqrt[3]{{\pi}^{3}}}{4}} \]

   7. associate-/r* 96.4%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(0.125, \pi \cdot f, \color{blue}{\frac{\frac{4}{\pi}}{f}}\right)\right)}{\frac{\sqrt[3]{{\pi}^{3}}}{4}} \]

   8. rem-cbrt-cube 96.7%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(0.125, \pi \cdot f, \frac{\frac{4}{\pi}}{f}\right)\right)}{\frac{\color{blue}{\pi}}{4}} \]

   9. div-inv 96.7%

      \[\leadsto -\frac{\log \left(\mathsf{fma}\left(0.125, \pi \cdot f, \frac{\frac{4}{\pi}}{f}\right)\right)}{\color{blue}{\pi \cdot \frac{1}{4}}} \]

   10. metadata-eval 96.7%

       \[\leadsto -\frac{\log \left(\mathsf{fma}\left(0.125, \pi \cdot f, \frac{\frac{4}{\pi}}{f}\right)\right)}{\pi \cdot \color{blue}{0.25}} \]

10. Applied egg-rr 96.7%

    \[\leadsto -\color{blue}{\frac{\log \left(\mathsf{fma}\left(0.125, \pi \cdot f, \frac{\frac{4}{\pi}}{f}\right)\right)}{\pi \cdot 0.25}} \]

11. Final simplification 96.7%

    \[\leadsto \frac{-\log \left(\mathsf{fma}\left(0.125, \pi \cdot f, \frac{\frac{4}{\pi}}{f}\right)\right)}{\pi \cdot 0.25} \]
12. Add Preprocessing

Alternative 7: 95.9% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (* -4.0 (- (log (/ 2.0 (* PI 0.5))) (log f))) PI))

C

double code(double f) {
	return (-4.0 * (log((2.0 / (((double) M_PI) * 0.5))) - log(f))) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (-4.0 * (Math.log((2.0 / (Math.PI * 0.5))) - Math.log(f))) / Math.PI;
}

Python

def code(f):
	return (-4.0 * (math.log((2.0 / (math.pi * 0.5))) - math.log(f))) / math.pi

Julia

function code(f)
	return Float64(Float64(-4.0 * Float64(log(Float64(2.0 / Float64(pi * 0.5))) - log(f))) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (-4.0 * (log((2.0 / (pi * 0.5))) - log(f))) / pi;
end

Wolfram

code[f_] := N[(N[(-4.0 * N[(N[Log[N[(2.0 / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - N[Log[f], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

3. Simplified 6.9%

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

5. Taylor expanded in f around 0 96.6%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi}} \]
6. Step-by-step derivation

   1. *-commutative 96.6%

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

   2. associate-*l/ 96.6%

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

   3. mul-1-neg 96.6%

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

   4. unsub-neg 96.6%

      \[\leadsto \frac{\color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) - \log f\right)} \cdot -4}{\pi} \]

   5. distribute-rgt-out-- 96.6%

      \[\leadsto \frac{\left(\log \left(\frac{2}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) - \log f\right) \cdot -4}{\pi} \]

   6. metadata-eval 96.6%

      \[\leadsto \frac{\left(\log \left(\frac{2}{\pi \cdot \color{blue}{0.5}}\right) - \log f\right) \cdot -4}{\pi} \]

7. Simplified 96.6%

   \[\leadsto \color{blue}{\frac{\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right) \cdot -4}{\pi}} \]

8. Final simplification 96.6%

   \[\leadsto \frac{-4 \cdot \left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right)}{\pi} \]
9. Add Preprocessing

Alternative 8: 95.8% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \log \left(0.125 \cdot \left(\pi \cdot f\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (log (+ (* 0.125 (* PI f)) (* 4.0 (/ 1.0 (* PI f))))) (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(((0.125 * (((double) M_PI) * f)) + (4.0 * (1.0 / (((double) M_PI) * f))))) * (-1.0 / (((double) M_PI) / 4.0));
}

Java

public static double code(double f) {
	return Math.log(((0.125 * (Math.PI * f)) + (4.0 * (1.0 / (Math.PI * f))))) * (-1.0 / (Math.PI / 4.0));
}

Python

def code(f):
	return math.log(((0.125 * (math.pi * f)) + (4.0 * (1.0 / (math.pi * f))))) * (-1.0 / (math.pi / 4.0))

Julia

function code(f)
	return Float64(log(Float64(Float64(0.125 * Float64(pi * f)) + Float64(4.0 * Float64(1.0 / Float64(pi * f))))) * Float64(-1.0 / Float64(pi / 4.0)))
end

MATLAB

function tmp = code(f)
	tmp = log(((0.125 * (pi * f)) + (4.0 * (1.0 / (pi * f))))) * (-1.0 / (pi / 4.0));
end

Wolfram

code[f_] := N[(N[Log[N[(N[(0.125 * N[(Pi * f), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(1.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in f around 0 96.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{f \cdot \left(0.25 \cdot \pi - -0.25 \cdot \pi\right)}}\right) \]
4. Step-by-step derivation

   1. distribute-rgt-out-- 96.6%

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

   2. metadata-eval 96.6%

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

5. Simplified 96.6%

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

6. Taylor expanded in f around 0 96.6%

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

7. Final simplification 96.6%

   \[\leadsto \log \left(0.125 \cdot \left(\pi \cdot f\right) + 4 \cdot \frac{1}{\pi \cdot f}\right) \cdot \frac{-1}{\frac{\pi}{4}} \]
8. Add Preprocessing

Alternative 9: 95.9% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \frac{-4 \cdot \mathsf{log1p}\left(-1 + \frac{\frac{2}{\pi \cdot 0.5}}{f}\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (* -4.0 (log1p (+ -1.0 (/ (/ 2.0 (* PI 0.5)) f)))) PI))

C

double code(double f) {
	return (-4.0 * log1p((-1.0 + ((2.0 / (((double) M_PI) * 0.5)) / f)))) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (-4.0 * Math.log1p((-1.0 + ((2.0 / (Math.PI * 0.5)) / f)))) / Math.PI;
}

Python

def code(f):
	return (-4.0 * math.log1p((-1.0 + ((2.0 / (math.pi * 0.5)) / f)))) / math.pi

Julia

function code(f)
	return Float64(Float64(-4.0 * log1p(Float64(-1.0 + Float64(Float64(2.0 / Float64(pi * 0.5)) / f)))) / pi)
end

Wolfram

code[f_] := N[(N[(-4.0 * N[Log[1 + N[(-1.0 + N[(N[(2.0 / N[(Pi * 0.5), $MachinePrecision]), $MachinePrecision] / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

3. Simplified 6.9%

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

5. Taylor expanded in f around 0 96.6%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi}} \]
6. Step-by-step derivation

   1. *-commutative 96.6%

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

   2. associate-*l/ 96.6%

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

   3. mul-1-neg 96.6%

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

   4. unsub-neg 96.6%

      \[\leadsto \frac{\color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) - \log f\right)} \cdot -4}{\pi} \]

   5. distribute-rgt-out-- 96.6%

      \[\leadsto \frac{\left(\log \left(\frac{2}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) - \log f\right) \cdot -4}{\pi} \]

   6. metadata-eval 96.6%

      \[\leadsto \frac{\left(\log \left(\frac{2}{\pi \cdot \color{blue}{0.5}}\right) - \log f\right) \cdot -4}{\pi} \]

7. Simplified 96.6%

   \[\leadsto \color{blue}{\frac{\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right) \cdot -4}{\pi}} \]
8. Step-by-step derivation

   1. log1p-expm1-u 96.6%

      \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right)\right)} \cdot -4}{\pi} \]

   2. expm1-udef 96.6%

      \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{e^{\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f} - 1}\right) \cdot -4}{\pi} \]

   3. diff-log 96.6%

      \[\leadsto \frac{\mathsf{log1p}\left(e^{\color{blue}{\log \left(\frac{\frac{2}{\pi \cdot 0.5}}{f}\right)}} - 1\right) \cdot -4}{\pi} \]

   4. add-exp-log 96.6%

      \[\leadsto \frac{\mathsf{log1p}\left(\color{blue}{\frac{\frac{2}{\pi \cdot 0.5}}{f}} - 1\right) \cdot -4}{\pi} \]

9. Applied egg-rr 96.6%

   \[\leadsto \frac{\color{blue}{\mathsf{log1p}\left(\frac{\frac{2}{\pi \cdot 0.5}}{f} - 1\right)} \cdot -4}{\pi} \]

10. Final simplification 96.6%

    \[\leadsto \frac{-4 \cdot \mathsf{log1p}\left(-1 + \frac{\frac{2}{\pi \cdot 0.5}}{f}\right)}{\pi} \]
11. Add Preprocessing

Alternative 10: 95.9% accurate, 4.9× speedup

Math

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

FPCore

(FPCore (f) :precision binary64 (* -4.0 (/ (log (/ 4.0 (* PI f))) PI)))

C

double code(double f) {
	return -4.0 * (log((4.0 / (((double) M_PI) * f))) / ((double) M_PI));
}

Java

public static double code(double f) {
	return -4.0 * (Math.log((4.0 / (Math.PI * f))) / Math.PI);
}

Python

def code(f):
	return -4.0 * (math.log((4.0 / (math.pi * f))) / math.pi)

Julia

function code(f)
	return Float64(-4.0 * Float64(log(Float64(4.0 / Float64(pi * f))) / pi))
end

MATLAB

function tmp = code(f)
	tmp = -4.0 * (log((4.0 / (pi * f))) / pi);
end

Wolfram

code[f_] := N[(-4.0 * N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

   \[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
2. Step-by-step derivation

   1. distribute-lft-neg-in 6.9%

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

   2. *-commutative 6.9%

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

3. Simplified 6.9%

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

5. Taylor expanded in f around 0 96.6%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) + -1 \cdot \log f}{\pi}} \]
6. Step-by-step derivation

   1. *-commutative 96.6%

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

   2. associate-*l/ 96.6%

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

   3. mul-1-neg 96.6%

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

   4. unsub-neg 96.6%

      \[\leadsto \frac{\color{blue}{\left(\log \left(\frac{2}{0.25 \cdot \pi - -0.25 \cdot \pi}\right) - \log f\right)} \cdot -4}{\pi} \]

   5. distribute-rgt-out-- 96.6%

      \[\leadsto \frac{\left(\log \left(\frac{2}{\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}}\right) - \log f\right) \cdot -4}{\pi} \]

   6. metadata-eval 96.6%

      \[\leadsto \frac{\left(\log \left(\frac{2}{\pi \cdot \color{blue}{0.5}}\right) - \log f\right) \cdot -4}{\pi} \]

7. Simplified 96.6%

   \[\leadsto \color{blue}{\frac{\left(\log \left(\frac{2}{\pi \cdot 0.5}\right) - \log f\right) \cdot -4}{\pi}} \]

8. Taylor expanded in f around 0 96.6%

   \[\leadsto \color{blue}{-4 \cdot \frac{\log \left(\frac{4}{\pi}\right) - \log f}{\pi}} \]
9. Step-by-step derivation

   1. log-div 96.6%

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

   2. associate-*r/ 96.6%

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

   3. log-div 96.6%

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

   4. remove-double-neg 96.6%

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

   5. mul-1-neg 96.6%

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

   6. log-rec 96.6%

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

   7. associate-*r/ 96.6%

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

   8. div-sub 96.6%

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

   9. log-rec 96.6%

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

   10. mul-1-neg 96.6%

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

   11. remove-double-neg 96.6%

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

   12. div-sub 96.6%

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

10. Simplified 96.6%

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

11. Final simplification 96.6%

    \[\leadsto -4 \cdot \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi} \]
12. Add Preprocessing

Alternative 11: 0.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{-\log 0}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* 4.0 (/ (- (log 0.0)) PI)))

C

double code(double f) {
	return 4.0 * (-log(0.0) / ((double) M_PI));
}

Java

public static double code(double f) {
	return 4.0 * (-Math.log(0.0) / Math.PI);
}

Python

def code(f):
	return 4.0 * (-math.log(0.0) / math.pi)

Julia

function code(f)
	return Float64(4.0 * Float64(Float64(-log(0.0)) / pi))
end

MATLAB

function tmp = code(f)
	tmp = 4.0 * (-log(0.0) / pi);
end

Wolfram

code[f_] := N[(4.0 * N[((-N[Log[0.0], $MachinePrecision]) / Pi), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.9%

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

3. Taylor expanded in f around 0 96.5%

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

4. Taylor expanded in f around inf 0.7%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log \left(-0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi} + 0.25 \cdot \frac{\pi}{0.25 \cdot \pi - -0.25 \cdot \pi}\right)}{\pi}} \]
5. Step-by-step derivation

   1. distribute-rgt-out 0.7%

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

   2. distribute-rgt-out-- 0.7%

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

   3. metadata-eval 0.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{\pi}{\pi \cdot \color{blue}{0.5}} \cdot \left(-0.25 + 0.25\right)\right)}{\pi} \]

   4. metadata-eval 0.7%

      \[\leadsto -4 \cdot \frac{\log \left(\frac{\pi}{\pi \cdot 0.5} \cdot \color{blue}{0}\right)}{\pi} \]

   5. mul0-rgt 0.7%

      \[\leadsto -4 \cdot \frac{\log \color{blue}{0}}{\pi} \]

6. Simplified 0.7%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0}{\pi}} \]

7. Final simplification 0.7%

   \[\leadsto 4 \cdot \frac{-\log 0}{\pi} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024013 
(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)))))))))