VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.6% → 96.8%

Time: 32.4s

Alternatives: 7

Speedup: 4.8×

• 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: 6.6% 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: 96.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{\left(f \cdot \pi\right) \cdot 0.25}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)}\right)}{\frac{\pi}{-4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (log
   (/
    (+ (exp (* -0.25 (* f PI))) (exp (* (* f PI) 0.25)))
    (fma
     f
     (* PI 0.5)
     (fma
      (pow f 3.0)
      (* (pow PI 3.0) 0.005208333333333333)
      (fma
       (pow f 5.0)
       (* (pow PI 5.0) 1.6276041666666666e-5)
       (* (pow f 7.0) (* (pow PI 7.0) 2.422030009920635e-8)))))))
  (/ PI -4.0)))

C

double code(double f) {
	return log(((exp((-0.25 * (f * ((double) M_PI)))) + exp(((f * ((double) M_PI)) * 0.25))) / fma(f, (((double) M_PI) * 0.5), fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), (pow(f, 7.0) * (pow(((double) M_PI), 7.0) * 2.422030009920635e-8))))))) / (((double) M_PI) / -4.0);
}

Julia

function code(f)
	return Float64(log(Float64(Float64(exp(Float64(-0.25 * Float64(f * pi))) + exp(Float64(Float64(f * pi) * 0.25))) / fma(f, Float64(pi * 0.5), fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), Float64((f ^ 7.0) * Float64((pi ^ 7.0) * 2.422030009920635e-8))))))) / Float64(pi / -4.0))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[Exp[N[(-0.25 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(f * N[(Pi * 0.5), $MachinePrecision] + N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $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[(Pi / -4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. distribute-lft-neg-in 7.7%

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

   2. distribute-neg-frac2 7.7%

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

   3. associate-*l/ 7.7%

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

   4. *-lft-identity 7.7%

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

3. Simplified 7.7%

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

5. Taylor expanded in f around inf 7.7%

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

6. Taylor expanded in f around 0 97.3%

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

   1. fma-define 97.3%

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

   2. distribute-rgt-out-- 97.3%

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

   3. metadata-eval 97.3%

      \[\leadsto \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + \left({f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right)}{\frac{\pi}{-4}} \]

   4. fma-define 97.3%

      \[\leadsto \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right)}\right)}{\frac{\pi}{-4}} \]

   5. distribute-rgt-out-- 97.3%

      \[\leadsto \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, \color{blue}{{\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right)}{\frac{\pi}{-4}} \]

   6. metadata-eval 97.3%

      \[\leadsto \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right)}{\frac{\pi}{-4}} \]

8. Simplified 97.3%

   \[\leadsto \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)}}\right)}{\frac{\pi}{-4}} \]

9. Final simplification 97.3%

   \[\leadsto \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{\left(f \cdot \pi\right) \cdot 0.25}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, {f}^{7} \cdot \left({\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}\right)\right)\right)\right)}\right)}{\frac{\pi}{-4}} \]
10. Add Preprocessing

Alternative 2: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{\left(f \cdot \pi\right) \cdot 0.25}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)}\right)}{\frac{\pi}{-4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/
  (log
   (/
    (+ (exp (* -0.25 (* f PI))) (exp (* (* f PI) 0.25)))
    (fma
     f
     (* PI 0.5)
     (fma
      (pow f 3.0)
      (* (pow PI 3.0) 0.005208333333333333)
      (* (pow f 5.0) (* (pow PI 5.0) 1.6276041666666666e-5))))))
  (/ PI -4.0)))

C

double code(double f) {
	return log(((exp((-0.25 * (f * ((double) M_PI)))) + exp(((f * ((double) M_PI)) * 0.25))) / fma(f, (((double) M_PI) * 0.5), fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), (pow(f, 5.0) * (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5)))))) / (((double) M_PI) / -4.0);
}

Julia

function code(f)
	return Float64(log(Float64(Float64(exp(Float64(-0.25 * Float64(f * pi))) + exp(Float64(Float64(f * pi) * 0.25))) / fma(f, Float64(pi * 0.5), fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), Float64((f ^ 5.0) * Float64((pi ^ 5.0) * 1.6276041666666666e-5)))))) / Float64(pi / -4.0))
end

Wolfram

code[f_] := N[(N[Log[N[(N[(N[Exp[N[(-0.25 * N[(f * Pi), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[Exp[N[(N[(f * Pi), $MachinePrecision] * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(f * N[(Pi * 0.5), $MachinePrecision] + N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $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[(Pi / -4.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. distribute-lft-neg-in 7.7%

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

   2. distribute-neg-frac2 7.7%

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

   3. associate-*l/ 7.7%

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

   4. *-lft-identity 7.7%

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

3. Simplified 7.7%

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

5. Taylor expanded in f around inf 7.7%

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

6. Taylor expanded in f around 0 97.3%

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

   1. fma-define 97.3%

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

   2. distribute-rgt-out-- 97.3%

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

   3. metadata-eval 97.3%

      \[\leadsto \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot \color{blue}{0.5}, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right)}{\frac{\pi}{-4}} \]

   4. fma-define 97.3%

      \[\leadsto \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)}\right)}\right)}{\frac{\pi}{-4}} \]

   5. distribute-rgt-out-- 97.3%

      \[\leadsto \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, \color{blue}{{\pi}^{3} \cdot \left(0.0026041666666666665 - -0.0026041666666666665\right)}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right)}{\frac{\pi}{-4}} \]

   6. metadata-eval 97.3%

      \[\leadsto \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot \color{blue}{0.005208333333333333}, {f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right)\right)\right)}\right)}{\frac{\pi}{-4}} \]

   7. distribute-rgt-out-- 97.3%

      \[\leadsto \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \color{blue}{\left({\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)\right)}\right)\right)}\right)}{\frac{\pi}{-4}} \]

   8. metadata-eval 97.3%

      \[\leadsto \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \left({\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}\right)\right)\right)}\right)}{\frac{\pi}{-4}} \]

8. Simplified 97.3%

   \[\leadsto \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{0.25 \cdot \left(f \cdot \pi\right)}}{\color{blue}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)}}\right)}{\frac{\pi}{-4}} \]

9. Final simplification 97.3%

   \[\leadsto \frac{\log \left(\frac{e^{-0.25 \cdot \left(f \cdot \pi\right)} + e^{\left(f \cdot \pi\right) \cdot 0.25}}{\mathsf{fma}\left(f, \pi \cdot 0.5, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {f}^{5} \cdot \left({\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)}\right)}{\frac{\pi}{-4}} \]
10. Add Preprocessing

Alternative 3: 96.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ 4 \cdot \frac{\log f - \log \left(\frac{4}{\pi}\right)}{\pi} - {f}^{2} \cdot \left(\pi \cdot 0.08333333333333333\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (-
  (* 4.0 (/ (- (log f) (log (/ 4.0 PI))) PI))
  (* (pow f 2.0) (* PI 0.08333333333333333))))

C

double code(double f) {
	return (4.0 * ((log(f) - log((4.0 / ((double) M_PI)))) / ((double) M_PI))) - (pow(f, 2.0) * (((double) M_PI) * 0.08333333333333333));
}

Java

public static double code(double f) {
	return (4.0 * ((Math.log(f) - Math.log((4.0 / Math.PI))) / Math.PI)) - (Math.pow(f, 2.0) * (Math.PI * 0.08333333333333333));
}

Python

def code(f):
	return (4.0 * ((math.log(f) - math.log((4.0 / math.pi))) / math.pi)) - (math.pow(f, 2.0) * (math.pi * 0.08333333333333333))

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(Float64(log(f) - log(Float64(4.0 / pi))) / pi)) - Float64((f ^ 2.0) * Float64(pi * 0.08333333333333333)))
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * ((log(f) - log((4.0 / pi))) / pi)) - ((f ^ 2.0) * (pi * 0.08333333333333333));
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[(N[Power[f, 2.0], $MachinePrecision] * N[(Pi * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in f around 0 96.6%

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

5. Simplified 96.6%

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

6. Taylor expanded in f around 0 96.7%

   \[\leadsto -\color{blue}{\left(4 \cdot \frac{\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f}{\pi} + {f}^{2} \cdot \left(-0.041666666666666664 \cdot \pi + 0.125 \cdot \pi\right)\right)} \]
7. Step-by-step derivation

   1. distribute-rgt-out 96.7%

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

   2. metadata-eval 96.7%

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

8. Applied egg-rr 96.7%

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

9. Final simplification 96.7%

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

Alternative 4: 96.1% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

   1. distribute-lft-neg-in 7.7%

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

   2. distribute-neg-frac2 7.7%

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

   3. associate-*l/ 7.7%

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

   4. *-lft-identity 7.7%

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

3. Simplified 7.7%

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

5. Taylor expanded in f around 0 95.7%

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

   1. *-commutative 95.7%

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

   2. associate-*l/ 95.7%

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

   3. associate-/l* 95.6%

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

   4. mul-1-neg 95.6%

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

   5. unsub-neg 95.6%

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

   6. distribute-rgt-out-- 95.6%

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

   7. metadata-eval 95.6%

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

   8. *-commutative 95.6%

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

   9. associate-/r* 95.6%

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

   10. metadata-eval 95.6%

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

7. Simplified 95.6%

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

8. Taylor expanded in f around 0 95.7%

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

   1. *-commutative 95.7%

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

10. Simplified 95.7%

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

11. Final simplification 95.7%

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

Alternative 5: 96.1% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.7%

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

   1. distribute-lft-neg-in 7.7%

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

   2. distribute-neg-frac2 7.7%

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

   3. associate-*l/ 7.7%

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

   4. *-lft-identity 7.7%

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

3. Simplified 7.7%

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

5. Taylor expanded in f around inf 7.7%

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

6. Taylor expanded in f around 0 95.7%

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

   1. associate-/r* 95.7%

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

   2. distribute-rgt-out-- 95.7%

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

   3. metadata-eval 95.7%

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

8. Simplified 95.7%

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

9. Final simplification 95.7%

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

Alternative 6: 3.1% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\log 0}{0.25}}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (/ (log 0.0) 0.25) PI))

C

double code(double f) {
	return (log(0.0) / 0.25) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (Math.log(0.0) / 0.25) / Math.PI;
}

Python

def code(f):
	return (math.log(0.0) / 0.25) / math.pi

Julia

function code(f)
	return Float64(Float64(log(0.0) / 0.25) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (log(0.0) / 0.25) / pi;
end

Wolfram

code[f_] := N[(N[(N[Log[0.0], $MachinePrecision] / 0.25), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. distribute-lft-neg-in 7.7%

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

   2. distribute-neg-frac2 7.7%

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

   3. associate-*l/ 7.7%

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

   4. *-lft-identity 7.7%

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

3. Simplified 7.7%

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

5. Taylor expanded in f around inf 7.7%

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

   1. log-div 7.7%

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

   2. div-sub 7.7%

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

7. Applied egg-rr 3.1%

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

   1. div-sub 3.1%

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

9. Simplified 3.1%

   \[\leadsto \color{blue}{\frac{\frac{\log 0}{0.25}}{\pi}} \]

10. Final simplification 3.1%

    \[\leadsto \frac{\frac{\log 0}{0.25}}{\pi} \]
11. Add Preprocessing

Alternative 7: 1.6% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ -\mathsf{log1p}\left(-0.020833333333333332\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (- (log1p -0.020833333333333332)))

C

double code(double f) {
	return -log1p(-0.020833333333333332);
}

Java

public static double code(double f) {
	return -Math.log1p(-0.020833333333333332);
}

Python

def code(f):
	return -math.log1p(-0.020833333333333332)

Julia

function code(f)
	return Float64(-log1p(-0.020833333333333332))
end

Wolfram

code[f_] := (-N[Log[1 + -0.020833333333333332], $MachinePrecision])

Derivation

1. Initial program 7.7%

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

3. Taylor expanded in f around 0 96.6%

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

5. Simplified 96.6%

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

7. Applied egg-rr 1.6%

   \[\leadsto -\color{blue}{\mathsf{log1p}\left(-0.020833333333333332\right)} \]

8. Final simplification 1.6%

   \[\leadsto -\mathsf{log1p}\left(-0.020833333333333332\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024044 
(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)))))))))