VandenBroeck and Keller, Equation (20)

Percentage Accurate: 6.8% → 96.3%

Time: 28.4s

Alternatives: 10

Speedup: 3.4×

• could not determine a ground truth

  1. f = -2.1922397537229744e+188

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: 6.8% 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: 96.3% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

2. Taylor expanded in f around 0 96.4%

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

   1. fma-def 96.4%

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

   2. distribute-rgt-out-- 96.4%

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

   3. metadata-eval 96.4%

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

   4. associate-+r+ 96.4%

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

   5. +-commutative 96.4%

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

4. Simplified 96.4%

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

   1. log-div 96.4%

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

   2. cosh-undef 96.4%

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

   3. associate-*l/ 96.4%

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

6. Applied egg-rr 96.4%

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

7. Final simplification 96.4%

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

Alternative 2: 96.3% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log
   (/
    2.0
    (/
     (fma
      (pow f 5.0)
      (* (pow PI 5.0) 1.6276041666666666e-5)
      (fma
       (pow f 3.0)
       (* (pow PI 3.0) 0.005208333333333333)
       (fma
        PI
        (* f 0.5)
        (* (pow PI 7.0) (* (pow f 7.0) 2.422030009920635e-8)))))
     (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(pow(f, 3.0), (pow(((double) M_PI), 3.0) * 0.005208333333333333), fma(((double) M_PI), (f * 0.5), (pow(((double) M_PI), 7.0) * (pow(f, 7.0) * 2.422030009920635e-8))))) / 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((f ^ 3.0), Float64((pi ^ 3.0) * 0.005208333333333333), fma(pi, Float64(f * 0.5), Float64((pi ^ 7.0) * Float64((f ^ 7.0) * 2.422030009920635e-8))))) / 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[Power[f, 3.0], $MachinePrecision] * N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(Pi * N[(f * 0.5), $MachinePrecision] + N[(N[Power[Pi, 7.0], $MachinePrecision] * N[(N[Power[f, 7.0], $MachinePrecision] * 2.422030009920635e-8), $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 6.4%

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

2. Taylor expanded in f around 0 96.4%

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

   1. fma-def 96.4%

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

   2. distribute-rgt-out-- 96.4%

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

   3. metadata-eval 96.4%

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

   4. associate-+r+ 96.4%

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

   5. +-commutative 96.4%

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

4. Simplified 96.4%

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

   1. log-div 96.4%

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

   2. cosh-undef 96.4%

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

   3. associate-*l/ 96.4%

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

6. Applied egg-rr 96.4%

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

   1. log-div 96.4%

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

   2. associate-/l* 96.4%

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

   3. associate-/l* 96.4%

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

8. Simplified 96.4%

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

9. Final simplification 96.4%

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

Alternative 3: 96.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

2. Taylor expanded in f around 0 96.2%

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

4. Simplified 96.1%

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

5. Final simplification 96.1%

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

Alternative 4: 96.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

2. Taylor expanded in f around 0 96.1%

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

4. Simplified 96.1%

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

5. Taylor expanded in f around 0 96.1%

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

   1. associate-/r* 96.1%

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

7. Simplified 96.1%

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

8. Final simplification 96.1%

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

Alternative 5: 96.0% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (f)
 :precision binary64
 (*
  (log (fma f (* PI 0.08333333333333333) (/ (/ 4.0 PI) f)))
  (/ -1.0 (/ PI 4.0))))

C

double code(double f) {
	return log(fma(f, (((double) M_PI) * 0.08333333333333333), ((4.0 / ((double) M_PI)) / f))) * (-1.0 / (((double) M_PI) / 4.0));
}

Julia

function code(f)
	return Float64(log(fma(f, Float64(pi * 0.08333333333333333), Float64(Float64(4.0 / pi) / f))) * Float64(-1.0 / Float64(pi / 4.0)))
end

Wolfram

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

Derivation

1. Initial program 6.4%

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

2. Taylor expanded in f around 0 96.1%

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

4. Simplified 96.1%

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

   1. fma-udef 96.1%

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

   2. *-commutative 96.1%

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

   3. associate-/r/ 96.1%

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

   4. associate-*r/ 96.1%

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

   5. metadata-eval 96.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \left(\pi \cdot 2\right) + \frac{-2}{\frac{\color{blue}{0.25}}{\pi}} \cdot 0.005208333333333333, \frac{\frac{4}{\pi}}{f}\right)\right) \]

6. Applied egg-rr 96.1%

   \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{0.0625 \cdot \left(\pi \cdot 2\right) + \frac{-2}{\frac{0.25}{\pi}} \cdot 0.005208333333333333}, \frac{\frac{4}{\pi}}{f}\right)\right) \]
7. Step-by-step derivation

   1. *-commutative 96.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, 0.0625 \cdot \color{blue}{\left(2 \cdot \pi\right)} + \frac{-2}{\frac{0.25}{\pi}} \cdot 0.005208333333333333, \frac{\frac{4}{\pi}}{f}\right)\right) \]

   2. associate-*r* 96.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\left(0.0625 \cdot 2\right) \cdot \pi} + \frac{-2}{\frac{0.25}{\pi}} \cdot 0.005208333333333333, \frac{\frac{4}{\pi}}{f}\right)\right) \]

   3. metadata-eval 96.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{0.125} \cdot \pi + \frac{-2}{\frac{0.25}{\pi}} \cdot 0.005208333333333333, \frac{\frac{4}{\pi}}{f}\right)\right) \]

   4. associate-*l/ 96.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, 0.125 \cdot \pi + \color{blue}{\frac{-2 \cdot 0.005208333333333333}{\frac{0.25}{\pi}}}, \frac{\frac{4}{\pi}}{f}\right)\right) \]

   5. associate-/r/ 96.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, 0.125 \cdot \pi + \color{blue}{\frac{-2 \cdot 0.005208333333333333}{0.25} \cdot \pi}, \frac{\frac{4}{\pi}}{f}\right)\right) \]

   6. metadata-eval 96.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, 0.125 \cdot \pi + \frac{\color{blue}{-0.010416666666666666}}{0.25} \cdot \pi, \frac{\frac{4}{\pi}}{f}\right)\right) \]

   7. metadata-eval 96.1%

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

   8. metadata-eval 96.1%

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

   9. cancel-sign-sub-inv 96.1%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{0.125 \cdot \pi - 0.041666666666666664 \cdot \pi}, \frac{\frac{4}{\pi}}{f}\right)\right) \]

   10. distribute-rgt-out-- 96.1%

       \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\mathsf{fma}\left(f, \color{blue}{\pi \cdot \left(0.125 - 0.041666666666666664\right)}, \frac{\frac{4}{\pi}}{f}\right)\right) \]

   11. metadata-eval 96.1%

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

8. Simplified 96.1%

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

9. Final simplification 96.1%

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

Alternative 6: 95.6% 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 6.4%

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

2. Taylor expanded in f around 0 95.9%

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

   1. associate-*r/ 95.9%

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

   2. associate-/l* 95.8%

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

   3. associate-/r/ 95.8%

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

   4. mul-1-neg 95.8%

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

   5. unsub-neg 95.8%

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

   6. distribute-rgt-out-- 95.8%

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

   7. *-commutative 95.8%

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

   8. associate-/r* 95.8%

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

   9. metadata-eval 95.8%

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

   10. metadata-eval 95.8%

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

4. Simplified 95.8%

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

5. Taylor expanded in f around 0 95.9%

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

6. Final simplification 95.9%

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

Alternative 7: 95.5% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

2. Taylor expanded in f around 0 95.7%

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

   1. associate-/r* 95.7%

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

   2. distribute-rgt-out-- 95.7%

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

   3. *-commutative 95.7%

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

   4. associate-/r* 95.7%

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

   5. metadata-eval 95.7%

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

   6. metadata-eval 95.7%

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

4. Simplified 95.7%

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

5. Taylor expanded in f around 0 95.9%

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

   1. associate-*r/ 95.9%

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

   2. neg-mul-1 95.9%

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

   3. log-rec 95.9%

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

   4. +-commutative 95.9%

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

   5. log-rec 95.9%

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

   6. sub-neg 95.9%

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

   7. log-div 95.8%

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

   8. associate-*l/ 95.7%

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

   9. associate-/l/ 95.7%

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

7. Simplified 95.7%

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

8. Final simplification 95.7%

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

Alternative 8: 95.6% accurate, 3.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 6.4%

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

2. Taylor expanded in f around 0 95.7%

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

   1. associate-/r* 95.7%

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

   2. distribute-rgt-out-- 95.7%

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

   3. *-commutative 95.7%

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

   4. associate-/r* 95.7%

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

   5. metadata-eval 95.7%

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

   6. metadata-eval 95.7%

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

4. Simplified 95.7%

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

   1. clear-num 95.7%

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

   2. diff-log 95.8%

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

   3. associate-*l/ 95.9%

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

   4. diff-log 95.8%

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

6. Applied egg-rr 95.8%

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

7. Final simplification 95.8%

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

Alternative 9: 13.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(-\left|\log 0.6666666666666666\right|\right)}{\pi} \end{array} \]

FPCore

(FPCore (f)
 :precision binary64
 (/ (* 4.0 (- (fabs (log 0.6666666666666666)))) PI))

C

double code(double f) {
	return (4.0 * -fabs(log(0.6666666666666666))) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (4.0 * -Math.abs(Math.log(0.6666666666666666))) / Math.PI;
}

Python

def code(f):
	return (4.0 * -math.fabs(math.log(0.6666666666666666))) / math.pi

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(-abs(log(0.6666666666666666)))) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * -abs(log(0.6666666666666666))) / pi;
end

Wolfram

code[f_] := N[(N[(4.0 * (-N[Abs[N[Log[0.6666666666666666], $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 6.4%

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

3. Applied egg-rr 1.6%

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

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.6666666666666666}{\pi}} \]
5. Step-by-step derivation

   1. associate-*r/ 1.6%

      \[\leadsto -\color{blue}{\frac{4 \cdot \log 0.6666666666666666}{\pi}} \]

6. Simplified 1.6%

   \[\leadsto -\color{blue}{\frac{4 \cdot \log 0.6666666666666666}{\pi}} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 0.0%

      \[\leadsto -\frac{4 \cdot \color{blue}{\left(\sqrt{\log 0.6666666666666666} \cdot \sqrt{\log 0.6666666666666666}\right)}}{\pi} \]

   2. sqrt-unprod 13.5%

      \[\leadsto -\frac{4 \cdot \color{blue}{\sqrt{\log 0.6666666666666666 \cdot \log 0.6666666666666666}}}{\pi} \]

   3. pow2 13.5%

      \[\leadsto -\frac{4 \cdot \sqrt{\color{blue}{{\log 0.6666666666666666}^{2}}}}{\pi} \]

8. Applied egg-rr 13.5%

   \[\leadsto -\frac{4 \cdot \color{blue}{\sqrt{{\log 0.6666666666666666}^{2}}}}{\pi} \]
9. Step-by-step derivation

   1. unpow2 13.5%

      \[\leadsto -\frac{4 \cdot \sqrt{\color{blue}{\log 0.6666666666666666 \cdot \log 0.6666666666666666}}}{\pi} \]

   2. rem-sqrt-square 13.5%

      \[\leadsto -\frac{4 \cdot \color{blue}{\left|\log 0.6666666666666666\right|}}{\pi} \]

10. Simplified 13.5%

    \[\leadsto -\frac{4 \cdot \color{blue}{\left|\log 0.6666666666666666\right|}}{\pi} \]

11. Final simplification 13.5%

    \[\leadsto \frac{4 \cdot \left(-\left|\log 0.6666666666666666\right|\right)}{\pi} \]

Alternative 10: 1.6% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{4 \cdot \left(-\log 0.6666666666666666\right)}{\pi} \end{array} \]

FPCore

(FPCore (f) :precision binary64 (/ (* 4.0 (- (log 0.6666666666666666))) PI))

C

double code(double f) {
	return (4.0 * -log(0.6666666666666666)) / ((double) M_PI);
}

Java

public static double code(double f) {
	return (4.0 * -Math.log(0.6666666666666666)) / Math.PI;
}

Python

def code(f):
	return (4.0 * -math.log(0.6666666666666666)) / math.pi

Julia

function code(f)
	return Float64(Float64(4.0 * Float64(-log(0.6666666666666666))) / pi)
end

MATLAB

function tmp = code(f)
	tmp = (4.0 * -log(0.6666666666666666)) / pi;
end

Wolfram

code[f_] := N[(N[(4.0 * (-N[Log[0.6666666666666666], $MachinePrecision])), $MachinePrecision] / Pi), $MachinePrecision]

Derivation

1. Initial program 6.4%

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

3. Applied egg-rr 1.6%

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

4. Taylor expanded in f around 0 1.6%

   \[\leadsto -\color{blue}{4 \cdot \frac{\log 0.6666666666666666}{\pi}} \]
5. Step-by-step derivation

   1. associate-*r/ 1.6%

      \[\leadsto -\color{blue}{\frac{4 \cdot \log 0.6666666666666666}{\pi}} \]

6. Simplified 1.6%

   \[\leadsto -\color{blue}{\frac{4 \cdot \log 0.6666666666666666}{\pi}} \]

7. Final simplification 1.6%

   \[\leadsto \frac{4 \cdot \left(-\log 0.6666666666666666\right)}{\pi} \]

Reproduce

herbie shell --seed 2023194 
(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)))))))))