VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.9% → 96.7%

Time: 30.6s

Alternatives: 11

Speedup: 3.4×

• could not determine a ground truth

  1. f = -7.074261952661113e+215

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: 96.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double f) {
	return log(((exp(((((double) M_PI) / 4.0) * f)) + exp(((((double) M_PI) / 4.0) * -f))) / fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), fma((((double) M_PI) * 0.5), f, fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), (pow(((double) M_PI), 7.0) * (2.422030009920635e-8 * pow(f, 7.0)))))))) * (-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(Float64(pi / 4.0) * Float64(-f)))) / fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), fma(Float64(pi * 0.5), f, fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), Float64((pi ^ 7.0) * Float64(2.422030009920635e-8 * (f ^ 7.0)))))))) * 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[(N[(Pi / 4.0), $MachinePrecision] * (-f)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(N[(Pi * 0.5), $MachinePrecision] * f + N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(N[Power[Pi, 7.0], $MachinePrecision] * N[(2.422030009920635e-8 * N[Power[f, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 94.4%

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

   1. fma-def 94.4%

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

   2. distribute-rgt-out-- 94.4%

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

   3. metadata-eval 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   4. fma-def 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{\mathsf{fma}\left(0.25 \cdot \pi - -0.25 \cdot \pi, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right)}\right) \]

   5. distribute-rgt-out-- 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   6. metadata-eval 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot \color{blue}{0.5}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   7. fma-def 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right)\right)}\right) \]

4. Simplified 94.4%

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

5. Final simplification 94.4%

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

Alternative 2: 96.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \log \left(\frac{2}{\frac{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {\pi}^{7} \cdot \left(2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}\right)\right)\right)\right)}{\cosh \left(\frac{\pi}{\frac{4}{f}}\right)}}\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    2.0
    (/
     (fma
      (pow f 5.0)
      (* (pow PI 5.0) 1.6276041666666666e-5)
      (fma
       (* PI 0.5)
       f
       (fma
        (pow f 3.0)
        (* (pow PI 3.0) 0.005208333333333333)
        (* (pow PI 7.0) (* 2.422030009920635e-8 (pow f 7.0))))))
     (cosh (/ PI (/ 4.0 f))))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log((2.0 / (fma(pow(f, 5.0), (pow(((double) M_PI), 5.0) * 1.6276041666666666e-5), fma((((double) M_PI) * 0.5), f, fma(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), (pow(((double) M_PI), 7.0) * (2.422030009920635e-8 * pow(f, 7.0)))))) / cosh((((double) M_PI) / (4.0 / f)))))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(Float64(2.0 / Float64(fma((f ^ 5.0), Float64((pi ^ 5.0) * 1.6276041666666666e-5), fma(Float64(pi * 0.5), f, fma((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), Float64((pi ^ 7.0) * Float64(2.422030009920635e-8 * (f ^ 7.0)))))) / cosh(Float64(pi / Float64(4.0 / f)))))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(2.0 / N[(N[(N[Power[f, 5.0], $MachinePrecision] * N[(N[Power[Pi, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision] + N[(N[(Pi * 0.5), $MachinePrecision] * f + N[(N[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(N[Power[Pi, 7.0], $MachinePrecision] * N[(2.422030009920635e-8 * N[Power[f, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Cosh[N[(Pi / N[(4.0 / f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 94.4%

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

   1. fma-def 94.4%

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

   2. distribute-rgt-out-- 94.4%

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

   3. metadata-eval 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   4. fma-def 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{\mathsf{fma}\left(0.25 \cdot \pi - -0.25 \cdot \pi, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right)}\right) \]

   5. distribute-rgt-out-- 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   6. metadata-eval 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot \color{blue}{0.5}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   7. fma-def 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right)\right)}\right) \]

4. Simplified 94.4%

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

   1. log-div 94.4%

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

   2. cosh-undef 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \color{blue}{\left(2 \cdot \cosh \left(\frac{\pi}{4} \cdot f\right)\right)} - \log \left(\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {\pi}^{7} \cdot \left(2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}\right)\right)\right)\right)\right)\right) \]

   3. associate-*l/ 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \left(\log \left(2 \cdot \cosh \color{blue}{\left(\frac{\pi \cdot f}{4}\right)}\right) - \log \left(\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {\pi}^{7} \cdot \left(2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}\right)\right)\right)\right)\right)\right) \]

6. Applied egg-rr 94.4%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)\right) - \log \left(\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {\pi}^{7} \cdot \left(2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}\right)\right)\right)\right)\right)\right)} \]
7. Step-by-step derivation

   1. log-div 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log \left(\frac{2 \cdot \cosh \left(\frac{\pi \cdot f}{4}\right)}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {\pi}^{7} \cdot \left(2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}\right)\right)\right)\right)}\right)} \]

   2. associate-/l* 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2}{\frac{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {\pi}^{7} \cdot \left(2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}\right)\right)\right)\right)}{\cosh \left(\frac{\pi \cdot f}{4}\right)}}\right)} \]

   3. associate-/l* 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{2}{\frac{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {\pi}^{7} \cdot \left(2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}\right)\right)\right)\right)}{\cosh \color{blue}{\left(\frac{\pi}{\frac{4}{f}}\right)}}}\right) \]

8. Simplified 94.4%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\log \left(\frac{2}{\frac{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {\pi}^{7} \cdot \left(2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}\right)\right)\right)\right)}{\cosh \left(\frac{\pi}{\frac{4}{f}}\right)}}\right)} \]

9. Final simplification 94.4%

   \[\leadsto \log \left(\frac{2}{\frac{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{3}, {\pi}^{3} \cdot 0.005208333333333333, {\pi}^{7} \cdot \left(2.422030009920635 \cdot 10^{-8} \cdot {f}^{7}\right)\right)\right)\right)}{\cosh \left(\frac{\pi}{\frac{4}{f}}\right)}}\right) \cdot \frac{-1}{\frac{\pi}{4}} \]

Alternative 3: 96.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{\pi \cdot 0.0625}{0.5}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (fma
    f
    (fma
     (* (/ (pow PI 3.0) (pow PI 2.0)) 0.020833333333333332)
     -2.0
     (/ (* PI 0.0625) 0.5))
    (/ 2.0 (* PI (* f 0.5)))))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(fma(f, fma(((pow(((double) M_PI), 3.0) / pow(((double) M_PI), 2.0)) * 0.020833333333333332), -2.0, ((((double) M_PI) * 0.0625) / 0.5)), (2.0 / (((double) M_PI) * (f * 0.5))))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(fma(f, fma(Float64(Float64((pi ^ 3.0) / (pi ^ 2.0)) * 0.020833333333333332), -2.0, Float64(Float64(pi * 0.0625) / 0.5)), Float64(2.0 / Float64(pi * Float64(f * 0.5))))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[Log[N[(f * N[(N[(N[(N[Power[Pi, 3.0], $MachinePrecision] / N[Power[Pi, 2.0], $MachinePrecision]), $MachinePrecision] * 0.020833333333333332), $MachinePrecision] * -2.0 + N[(N[(Pi * 0.0625), $MachinePrecision] / 0.5), $MachinePrecision]), $MachinePrecision] + N[(2.0 / N[(Pi * N[(f * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 94.0%

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

4. Simplified 94.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{0.0625 \cdot \pi}{0.5} \cdot 1\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right)} \]

5. Final simplification 94.0%

   \[\leadsto \log \left(\mathsf{fma}\left(f, \mathsf{fma}\left(\frac{{\pi}^{3}}{{\pi}^{2}} \cdot 0.020833333333333332, -2, \frac{\pi \cdot 0.0625}{0.5}\right), \frac{2}{\pi \cdot \left(f \cdot 0.5\right)}\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \]

Alternative 4: 96.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \left(\log \left(\frac{4}{\pi \cdot f}\right) + \mathsf{fma}\left(0.5, f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right), f \cdot 0\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (*
  (+
   (log (/ 4.0 (* PI f)))
   (fma
    0.5
    (* f (* f (fma 0.5 (* (pow PI 2.0) 0.08333333333333333) 0.0)))
    (* f 0.0)))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return (log((4.0 / (((double) M_PI) * f))) + fma(0.5, (f * (f * fma(0.5, (pow(((double) M_PI), 2.0) * 0.08333333333333333), 0.0))), (f * 0.0))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(Float64(log(Float64(4.0 / Float64(pi * f))) + fma(0.5, Float64(f * Float64(f * fma(0.5, Float64((pi ^ 2.0) * 0.08333333333333333), 0.0))), Float64(f * 0.0))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

code[f_] := N[(N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(0.5 * N[(f * N[(f * N[(0.5 * N[(N[Power[Pi, 2.0], $MachinePrecision] * 0.08333333333333333), $MachinePrecision] + 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(f * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 94.4%

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

   1. fma-def 94.4%

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

   2. distribute-rgt-out-- 94.4%

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

   3. metadata-eval 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   4. fma-def 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{\mathsf{fma}\left(0.25 \cdot \pi - -0.25 \cdot \pi, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right)}\right) \]

   5. distribute-rgt-out-- 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   6. metadata-eval 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot \color{blue}{0.5}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   7. fma-def 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right)\right)}\right) \]

4. Simplified 94.4%

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

5. Taylor expanded in f around 0 94.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + \left(\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f\right)\right)\right)} \]

7. Simplified 94.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{4}{f \cdot \pi}\right) + \mathsf{fma}\left(0.5, f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right), f \cdot 0\right)\right)} \]

8. Final simplification 94.0%

   \[\leadsto \left(\log \left(\frac{4}{\pi \cdot f}\right) + \mathsf{fma}\left(0.5, f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right), f \cdot 0\right)\right) \cdot \frac{-1}{\frac{\pi}{4}} \]

Alternative 5: 96.4% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

function code(f)
	return Float64(-fma(0.08333333333333333, Float64(pi * Float64(f * f)), Float64(log(Float64(4.0 / Float64(pi * f))) * Float64(4.0 / pi))))
end

Wolfram

code[f_] := (-N[(0.08333333333333333 * N[(Pi * N[(f * f), $MachinePrecision]), $MachinePrecision] + N[(N[Log[N[(4.0 / N[(Pi * f), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 94.4%

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

   1. fma-def 94.4%

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

   2. distribute-rgt-out-- 94.4%

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

   3. metadata-eval 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   4. fma-def 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{\mathsf{fma}\left(0.25 \cdot \pi - -0.25 \cdot \pi, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right)}\right) \]

   5. distribute-rgt-out-- 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   6. metadata-eval 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot \color{blue}{0.5}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   7. fma-def 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right)\right)}\right) \]

4. Simplified 94.4%

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

5. Taylor expanded in f around 0 94.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + \left(\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f\right)\right)\right)} \]

7. Simplified 94.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{4}{f \cdot \pi}\right) + \mathsf{fma}\left(0.5, f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right), f \cdot 0\right)\right)} \]

8. Taylor expanded in f around 0 94.2%

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

   1. fma-def 94.2%

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

   2. *-commutative 94.2%

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

   3. unpow2 94.2%

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

   4. associate-*r/ 94.2%

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

   5. neg-mul-1 94.2%

      \[\leadsto -\mathsf{fma}\left(0.08333333333333333, \pi \cdot \left(f \cdot f\right), \frac{4 \cdot \left(\color{blue}{\left(-\log f\right)} + \log \left(\frac{4}{\pi}\right)\right)}{\pi}\right) \]

   6. log-rec 94.2%

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

   7. +-commutative 94.2%

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

   8. log-rec 94.2%

      \[\leadsto -\mathsf{fma}\left(0.08333333333333333, \pi \cdot \left(f \cdot f\right), \frac{4 \cdot \left(\log \left(\frac{4}{\pi}\right) + \color{blue}{\left(-\log f\right)}\right)}{\pi}\right) \]

   9. unsub-neg 94.2%

      \[\leadsto -\mathsf{fma}\left(0.08333333333333333, \pi \cdot \left(f \cdot f\right), \frac{4 \cdot \color{blue}{\left(\log \left(\frac{4}{\pi}\right) - \log f\right)}}{\pi}\right) \]

   10. log-div 94.1%

       \[\leadsto -\mathsf{fma}\left(0.08333333333333333, \pi \cdot \left(f \cdot f\right), \frac{4 \cdot \color{blue}{\log \left(\frac{\frac{4}{\pi}}{f}\right)}}{\pi}\right) \]

   11. associate-/r* 94.1%

       \[\leadsto -\mathsf{fma}\left(0.08333333333333333, \pi \cdot \left(f \cdot f\right), \frac{4 \cdot \log \color{blue}{\left(\frac{4}{\pi \cdot f}\right)}}{\pi}\right) \]

   12. associate-*l/ 94.0%

       \[\leadsto -\mathsf{fma}\left(0.08333333333333333, \pi \cdot \left(f \cdot f\right), \color{blue}{\frac{4}{\pi} \cdot \log \left(\frac{4}{\pi \cdot f}\right)}\right) \]

10. Simplified 94.0%

    \[\leadsto -\color{blue}{\mathsf{fma}\left(0.08333333333333333, \pi \cdot \left(f \cdot f\right), \frac{4}{\pi} \cdot \log \left(\frac{4}{f \cdot \pi}\right)\right)} \]

11. Final simplification 94.0%

    \[\leadsto -\mathsf{fma}\left(0.08333333333333333, \pi \cdot \left(f \cdot f\right), \log \left(\frac{4}{\pi \cdot f}\right) \cdot \frac{4}{\pi}\right) \]

Alternative 6: 95.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 93.8%

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

   1. distribute-rgt-out-- 93.8%

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

   2. metadata-eval 93.8%

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

4. Simplified 93.8%

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

5. Taylor expanded in f around 0 93.9%

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

   1. associate-*r/ 93.9%

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

   2. metadata-eval 93.9%

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

7. Simplified 93.9%

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

8. Final simplification 93.9%

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

Alternative 7: 95.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 93.6%

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

   1. distribute-rgt-out-- 93.6%

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

   2. metadata-eval 93.6%

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

4. Simplified 93.6%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{\left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)} \]
5. Step-by-step derivation

   1. expm1-log1p-u 92.3%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)\right)\right)} \]

   2. associate-*l/ 92.3%

      \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{1 \cdot \log \left(\frac{2}{\left(\pi \cdot 0.5\right) \cdot f}\right)}{\frac{\pi}{4}}}\right)\right) \]

   3. *-un-lft-identity 92.3%

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

   4. associate-*l* 92.3%

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

   5. div-inv 92.3%

      \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)}\right)}{\color{blue}{\pi \cdot \frac{1}{4}}}\right)\right) \]

   6. metadata-eval 92.3%

      \[\leadsto -\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)}\right)}{\pi \cdot \color{blue}{0.25}}\right)\right) \]

6. Applied egg-rr 92.3%

   \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\log \left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)}\right)}{\pi \cdot 0.25}\right)\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 93.7%

      \[\leadsto -\color{blue}{\frac{\log \left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)}\right)}{\pi \cdot 0.25}} \]

   2. add-sqr-sqrt 93.1%

      \[\leadsto -\color{blue}{\sqrt{\frac{\log \left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)}\right)}{\pi \cdot 0.25}} \cdot \sqrt{\frac{\log \left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)}\right)}{\pi \cdot 0.25}}} \]

   3. sqrt-unprod 94.0%

      \[\leadsto -\color{blue}{\sqrt{\frac{\log \left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)}\right)}{\pi \cdot 0.25} \cdot \frac{\log \left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)}\right)}{\pi \cdot 0.25}}} \]

   4. pow2 94.0%

      \[\leadsto -\sqrt{\color{blue}{{\left(\frac{\log \left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)}\right)}{\pi \cdot 0.25}\right)}^{2}}} \]

   5. *-un-lft-identity 94.0%

      \[\leadsto -\sqrt{{\left(\frac{\color{blue}{1 \cdot \log \left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)}\right)}}{\pi \cdot 0.25}\right)}^{2}} \]

   6. *-commutative 94.0%

      \[\leadsto -\sqrt{{\left(\frac{1 \cdot \log \left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)}\right)}{\color{blue}{0.25 \cdot \pi}}\right)}^{2}} \]

   7. times-frac 94.0%

      \[\leadsto -\sqrt{{\color{blue}{\left(\frac{1}{0.25} \cdot \frac{\log \left(\frac{2}{\pi \cdot \left(0.5 \cdot f\right)}\right)}{\pi}\right)}}^{2}} \]

   8. metadata-eval 94.0%

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

   9. *-commutative 94.0%

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

8. Applied egg-rr 94.0%

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

   1. unpow2 94.0%

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

   2. rem-sqrt-square 94.0%

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

   3. associate-*r/ 94.0%

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

   4. associate-*l/ 93.9%

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

   5. associate-/r* 93.9%

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

   6. *-rgt-identity 93.9%

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

   7. associate-*r/ 93.9%

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

   8. *-commutative 93.9%

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

   9. associate-/r* 93.9%

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

   10. metadata-eval 93.9%

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

10. Simplified 93.9%

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

11. Final simplification 93.9%

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

Alternative 8: 96.0% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 93.6%

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

   1. distribute-rgt-out-- 93.6%

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

   2. metadata-eval 93.6%

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

4. Simplified 93.6%

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

5. Taylor expanded in f around 0 93.8%

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

   1. neg-mul-1 93.8%

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

   2. log-rec 93.8%

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

   3. +-commutative 93.8%

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

   4. log-rec 93.8%

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

   5. sub-neg 93.8%

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

7. Simplified 93.8%

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

8. Final simplification 93.8%

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

Alternative 9: 96.0% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 93.6%

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

   1. distribute-rgt-out-- 93.6%

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

   2. metadata-eval 93.6%

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

4. Simplified 93.6%

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

5. Taylor expanded in f around 0 93.8%

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

   1. neg-mul-1 93.8%

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

   2. log-rec 93.8%

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

   3. +-commutative 93.8%

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

   4. log-rec 93.8%

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

   5. sub-neg 93.8%

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

   6. metadata-eval 93.8%

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

   7. associate-/r* 93.8%

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

   8. *-commutative 93.8%

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

   9. log-div 93.8%

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

   10. associate--l- 93.8%

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

   11. log-prod 93.7%

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

   12. associate-*r* 93.7%

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

7. Simplified 93.7%

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

8. Final simplification 93.7%

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

Alternative 10: 14.3% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \left|\frac{\log 0.07407407407407407}{\pi}\right| \cdot \left(-4\right) \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (* (fabs (/ (log 0.07407407407407407) PI)) (- 4.0)))

C

double code(double f) {
	return fabs((log(0.07407407407407407) / ((double) M_PI))) * -4.0;
}

Java

public static double code(double f) {
	return Math.abs((Math.log(0.07407407407407407) / Math.PI)) * -4.0;
}

Python

def code(f):
	return math.fabs((math.log(0.07407407407407407) / math.pi)) * -4.0

Julia

function code(f)
	return Float64(abs(Float64(log(0.07407407407407407) / pi)) * Float64(-4.0))
end

MATLAB

function tmp = code(f)
	tmp = abs((log(0.07407407407407407) / pi)) * -4.0;
end

Wolfram

code[f_] := N[(N[Abs[N[(N[Log[0.07407407407407407], $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision] * (-4.0)), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

3. Applied egg-rr 1.7%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{27}}\right) \]

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.07407407407407407}{\pi}} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 0.0%

      \[\leadsto -4 \cdot \color{blue}{\left(\sqrt{\frac{\log 0.07407407407407407}{\pi}} \cdot \sqrt{\frac{\log 0.07407407407407407}{\pi}}\right)} \]

   2. sqrt-unprod 14.1%

      \[\leadsto -4 \cdot \color{blue}{\sqrt{\frac{\log 0.07407407407407407}{\pi} \cdot \frac{\log 0.07407407407407407}{\pi}}} \]

   3. pow2 14.1%

      \[\leadsto -4 \cdot \sqrt{\color{blue}{{\left(\frac{\log 0.07407407407407407}{\pi}\right)}^{2}}} \]

6. Applied egg-rr 14.1%

   \[\leadsto -4 \cdot \color{blue}{\sqrt{{\left(\frac{\log 0.07407407407407407}{\pi}\right)}^{2}}} \]
7. Step-by-step derivation

   1. unpow2 14.1%

      \[\leadsto -4 \cdot \sqrt{\color{blue}{\frac{\log 0.07407407407407407}{\pi} \cdot \frac{\log 0.07407407407407407}{\pi}}} \]

   2. rem-sqrt-square 14.1%

      \[\leadsto -4 \cdot \color{blue}{\left|\frac{\log 0.07407407407407407}{\pi}\right|} \]

8. Simplified 14.1%

   \[\leadsto -4 \cdot \color{blue}{\left|\frac{\log 0.07407407407407407}{\pi}\right|} \]

9. Final simplification 14.1%

   \[\leadsto \left|\frac{\log 0.07407407407407407}{\pi}\right| \cdot \left(-4\right) \]

Alternative 11: 4.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ 0.08333333333333333 \cdot \left(\pi \cdot \left(f \cdot \left(-f\right)\right)\right) \end{array} \]

FPCore

(FPCore (f) :precision binary64 (* 0.08333333333333333 (* PI (* f (- f)))))

C

double code(double f) {
	return 0.08333333333333333 * (((double) M_PI) * (f * -f));
}

Java

public static double code(double f) {
	return 0.08333333333333333 * (Math.PI * (f * -f));
}

Python

def code(f):
	return 0.08333333333333333 * (math.pi * (f * -f))

Julia

function code(f)
	return Float64(0.08333333333333333 * Float64(pi * Float64(f * Float64(-f))))
end

MATLAB

function tmp = code(f)
	tmp = 0.08333333333333333 * (pi * (f * -f));
end

Wolfram

code[f_] := N[(0.08333333333333333 * N[(Pi * N[(f * (-f)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.1%

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

2. Taylor expanded in f around 0 94.4%

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

   1. fma-def 94.4%

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

   2. distribute-rgt-out-- 94.4%

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

   3. metadata-eval 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   4. fma-def 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{\mathsf{fma}\left(0.25 \cdot \pi - -0.25 \cdot \pi, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right)}\right) \]

   5. distribute-rgt-out-- 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   6. metadata-eval 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot \color{blue}{0.5}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   7. fma-def 94.4%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \color{blue}{\mathsf{fma}\left({f}^{3}, 0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}, {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right)\right)}\right) \]

4. Simplified 94.4%

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

5. Taylor expanded in f around 0 94.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(0.5 \cdot \left({f}^{2} \cdot \left(-0.25 \cdot {\left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)}^{2} + 0.5 \cdot \left(\left(0.125 \cdot \pi - 0.041666666666666664 \cdot \pi\right) \cdot \pi\right)\right)\right) + \left(0.5 \cdot \left(f \cdot \left(-0.25 \cdot \pi + 0.25 \cdot \pi\right)\right) + \left(\log \left(\frac{4}{\pi}\right) + -1 \cdot \log f\right)\right)\right)} \]

7. Simplified 94.0%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \color{blue}{\left(\log \left(\frac{4}{f \cdot \pi}\right) + \mathsf{fma}\left(0.5, f \cdot \left(f \cdot \mathsf{fma}\left(0.5, {\pi}^{2} \cdot 0.08333333333333333, 0\right)\right), f \cdot 0\right)\right)} \]

8. Taylor expanded in f around inf 4.3%

   \[\leadsto -\color{blue}{0.08333333333333333 \cdot \left({f}^{2} \cdot \pi\right)} \]
9. Step-by-step derivation

   1. *-commutative 4.3%

      \[\leadsto -0.08333333333333333 \cdot \color{blue}{\left(\pi \cdot {f}^{2}\right)} \]

   2. unpow2 4.3%

      \[\leadsto -0.08333333333333333 \cdot \left(\pi \cdot \color{blue}{\left(f \cdot f\right)}\right) \]

10. Simplified 4.3%

    \[\leadsto -\color{blue}{0.08333333333333333 \cdot \left(\pi \cdot \left(f \cdot f\right)\right)} \]

11. Final simplification 4.3%

    \[\leadsto 0.08333333333333333 \cdot \left(\pi \cdot \left(f \cdot \left(-f\right)\right)\right) \]

Reproduce

herbie shell --seed 2023238 
(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)))))))))