VandenBroeck and Keller, Equation (20)

Percentage Accurate: 3.5% → 99.4%

Time: 20.6s

Precision: binary64

Cost: 130564

• Unsound rule application detected in e-graph. Results may not be sound.

Math

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

↓

\[\begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ \mathbf{if}\;t_0 \leq 200:\\ \;\;\;\;\log \left(\frac{e^{t_0} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{7}, {\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}, \left({\pi}^{3} \cdot 0.005208333333333333\right) \cdot {f}^{3}\right)\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]

FPCore

(FPCore (f)
 :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)))))))))

↓

(FPCore (f)
 :precision binary64
 (let* ((t_0 (* (/ PI 4.0) f)))
   (if (<= t_0 200.0)
     (*
      (log
       (/
        (+ (exp t_0) (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 7.0)
           (* (pow PI 7.0) 2.422030009920635e-8)
           (* (* (pow PI 3.0) 0.005208333333333333) (pow f 3.0)))))))
      (/ -1.0 (/ PI 4.0)))
     (* (/ 1.0 (/ PI 4.0)) 0.0))))

C

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

↓

double code(double f) {
	double t_0 = (((double) M_PI) / 4.0) * f;
	double tmp;
	if (t_0 <= 200.0) {
		tmp = log(((exp(t_0) + 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, 7.0), (pow(((double) M_PI), 7.0) * 2.422030009920635e-8), ((pow(((double) M_PI), 3.0) * 0.005208333333333333) * pow(f, 3.0))))))) * (-1.0 / (((double) M_PI) / 4.0));
	} else {
		tmp = (1.0 / (((double) M_PI) / 4.0)) * 0.0;
	}
	return tmp;
}

Julia

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

↓

function code(f)
	t_0 = Float64(Float64(pi / 4.0) * f)
	tmp = 0.0
	if (t_0 <= 200.0)
		tmp = Float64(log(Float64(Float64(exp(t_0) + 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 ^ 7.0), Float64((pi ^ 7.0) * 2.422030009920635e-8), Float64(Float64((pi ^ 3.0) * 0.005208333333333333) * (f ^ 3.0))))))) * Float64(-1.0 / Float64(pi / 4.0)));
	else
		tmp = Float64(Float64(1.0 / Float64(pi / 4.0)) * 0.0);
	end
	return tmp
end

Wolfram

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

↓

code[f_] := Block[{t$95$0 = N[(N[(Pi / 4.0), $MachinePrecision] * f), $MachinePrecision]}, If[LessEqual[t$95$0, 200.0], N[(N[Log[N[(N[(N[Exp[t$95$0], $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, 7.0], $MachinePrecision] * N[(N[Power[Pi, 7.0], $MachinePrecision] * 2.422030009920635e-8), $MachinePrecision] + N[(N[(N[Power[Pi, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] * N[Power[f, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(Pi / 4.0), $MachinePrecision]), $MachinePrecision] * 0.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (PI.f64) 4) f) < 200

   1. Initial program 6.6%

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

   2. Taylor expanded in f around 0 98.1%

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

   3. Simplified 98.1%

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

      [Start] 98.1%	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{{f}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}\right) + \left(\left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]
      fma-def [=>] 98.1%	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\color{blue}{\mathsf{fma}\left({f}^{5}, 8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5} - -8.138020833333333 \cdot 10^{-6} \cdot {\pi}^{5}, \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}}\right) \]
      distribute-rgt-out-- [=>] 98.1%	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, \color{blue}{{\pi}^{5} \cdot \left(8.138020833333333 \cdot 10^{-6} - -8.138020833333333 \cdot 10^{-6}\right)}, \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]
      metadata-eval [=>] 98.1%	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot \color{blue}{1.6276041666666666 \cdot 10^{-5}}, \left(0.25 \cdot \pi - -0.25 \cdot \pi\right) \cdot f + \left({f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]
      fma-def [=>] 98.1%	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \color{blue}{\mathsf{fma}\left(0.25 \cdot \pi - -0.25 \cdot \pi, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)}\right)}\right) \]
      distribute-rgt-out-- [=>] 98.1%	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\color{blue}{\pi \cdot \left(0.25 - -0.25\right)}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]
      metadata-eval [=>] 98.1%	\[ -\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot \color{blue}{0.5}, f, {f}^{3} \cdot \left(0.0026041666666666665 \cdot {\pi}^{3} - -0.0026041666666666665 \cdot {\pi}^{3}\right) + {f}^{7} \cdot \left(1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7} - -1.2110150049603175 \cdot 10^{-8} \cdot {\pi}^{7}\right)\right)\right)}\right) \]

   if 200 < (*.f64 (/.f64 (PI.f64) 4) f)

   1. Initial program 0.7%

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

   2. Applied egg-rr 100.0%

      \[\leadsto -\frac{1}{\frac{\pi}{4}} \cdot \log \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 200:\\ \;\;\;\;\log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{7}, {\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}, \left({\pi}^{3} \cdot 0.005208333333333333\right) \cdot {f}^{3}\right)\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	130564

\[\begin{array}{l} t_0 := \frac{\pi}{4} \cdot f\\ \mathbf{if}\;t_0 \leq 200:\\ \;\;\;\;\log \left(\frac{e^{t_0} + e^{\frac{\pi}{4} \cdot \left(-f\right)}}{\mathsf{fma}\left({f}^{5}, {\pi}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}, \mathsf{fma}\left(\pi \cdot 0.5, f, \mathsf{fma}\left({f}^{7}, {\pi}^{7} \cdot 2.422030009920635 \cdot 10^{-8}, \left({\pi}^{3} \cdot 0.005208333333333333\right) \cdot {f}^{3}\right)\right)\right)}\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]

Alternative 2
Accuracy	99.2%
Cost	39428

\[\begin{array}{l} \mathbf{if}\;\frac{\pi}{4} \cdot f \leq 200:\\ \;\;\;\;\frac{-\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{4}{f}}{\pi}\right)\right)}{\pi \cdot 0.25}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]

Alternative 3
Accuracy	99.2%
Cost	32836

\[\begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{4}{f}}{\pi}\right)\right) \cdot \frac{-4}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]

Alternative 4
Accuracy	99.2%
Cost	26884

\[\begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\log \left(\left(\frac{\frac{4}{f}}{\pi} + \left(1 + f \cdot \left(\pi \cdot 0.08333333333333333\right)\right)\right) + -1\right) \cdot \frac{-1}{\frac{\pi}{4}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]

Alternative 5
Accuracy	98.9%
Cost	19844

\[\begin{array}{l} \mathbf{if}\;f \leq 1.3:\\ \;\;\;\;\frac{4}{\pi} \cdot \left(-\log \left(\frac{\frac{4}{f}}{\pi}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]

Alternative 6
Accuracy	99.0%
Cost	19844

\[\begin{array}{l} \mathbf{if}\;f \leq 1.3:\\ \;\;\;\;\frac{\log \left(\frac{4}{\pi \cdot f}\right) \cdot \left(-4\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]

Alternative 7
Accuracy	58.1%
Cost	13252

\[\begin{array}{l} \mathbf{if}\;f \leq 225:\\ \;\;\;\;\frac{4 \cdot \left(-\log 262144\right)}{\pi}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\pi}{4}} \cdot 0\\ \end{array} \]

Alternative 8
Accuracy	9.3%
Cost	13120

\[\frac{4 \cdot \left(-\log 262144\right)}{\pi} \]

Reproduce

herbie shell --seed 2023271 
(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)))))))))