Average Error: 61.5 → 2.3
Time: 18.5s
Precision: binary64
Cost: 110464
\[-\frac{1}{\frac{\pi}{4}} \cdot \log \left(\frac{e^{\frac{\pi}{4} \cdot f} + e^{-\frac{\pi}{4} \cdot f}}{e^{\frac{\pi}{4} \cdot f} - e^{-\frac{\pi}{4} \cdot f}}\right) \]
↓
\[\begin{array}{l} t_0 := e^{\frac{\pi}{4}}\\ \log \left(\frac{{t_0}^{f} + {t_0}^{\left(-f\right)}}{\mathsf{fma}\left({\pi}^{3}, {f}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left({f}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)}\right) \cdot \frac{-4}{\pi} \end{array} \]
(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 (exp (/ PI 4.0))))
   (*
    (log
     (/
      (+ (pow t_0 f) (pow t_0 (- f)))
      (fma
       (pow PI 3.0)
       (* (pow f 3.0) 0.005208333333333333)
       (fma
        PI
        (* f 0.5)
        (* (pow PI 5.0) (* (pow f 5.0) 1.6276041666666666e-5))))))
    (/ -4.0 PI))))
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 = exp((((double) M_PI) / 4.0));
	return log(((pow(t_0, f) + pow(t_0, -f)) / fma(pow(((double) M_PI), 3.0), (pow(f, 3.0) * 0.005208333333333333), fma(((double) M_PI), (f * 0.5), (pow(((double) M_PI), 5.0) * (pow(f, 5.0) * 1.6276041666666666e-5)))))) * (-4.0 / ((double) M_PI));
}

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 = exp(Float64(pi / 4.0))
	return Float64(log(Float64(Float64((t_0 ^ f) + (t_0 ^ Float64(-f))) / fma((pi ^ 3.0), Float64((f ^ 3.0) * 0.005208333333333333), fma(pi, Float64(f * 0.5), Float64((pi ^ 5.0) * Float64((f ^ 5.0) * 1.6276041666666666e-5)))))) * Float64(-4.0 / pi))
end

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[Exp[N[(Pi / 4.0), $MachinePrecision]], $MachinePrecision]}, N[(N[Log[N[(N[(N[Power[t$95$0, f], $MachinePrecision] + N[Power[t$95$0, (-f)], $MachinePrecision]), $MachinePrecision] / N[(N[Power[Pi, 3.0], $MachinePrecision] * N[(N[Power[f, 3.0], $MachinePrecision] * 0.005208333333333333), $MachinePrecision] + N[(Pi * N[(f * 0.5), $MachinePrecision] + N[(N[Power[Pi, 5.0], $MachinePrecision] * N[(N[Power[f, 5.0], $MachinePrecision] * 1.6276041666666666e-5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-4.0 / Pi), $MachinePrecision]), $MachinePrecision]]

-\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 := e^{\frac{\pi}{4}}\\
\log \left(\frac{{t_0}^{f} + {t_0}^{\left(-f\right)}}{\mathsf{fma}\left({\pi}^{3}, {f}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left({f}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)}\right) \cdot \frac{-4}{\pi}
\end{array}

Error

Derivation

  1. Initial program 61.5

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

  2. Simplified61.5

    \[\leadsto \color{blue}{\log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}{{\left(e^{\frac{\pi}{4}}\right)}^{f} - {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}\right) \cdot \frac{-4}{\pi}} \]
    Proof
    (*.f64 (log.f64 (/.f64 (+.f64 (pow.f64 (exp.f64 (/.f64 (PI.f64) 4)) f) (pow.f64 (exp.f64 (/.f64 (PI.f64) 4)) (neg.f64 f))) (-.f64 (pow.f64 (exp.f64 (/.f64 (PI.f64) 4)) f) (pow.f64 (exp.f64 (/.f64 (PI.f64) 4)) (neg.f64 f))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
    (*.f64 (log.f64 (/.f64 (+.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f))) (pow.f64 (exp.f64 (/.f64 (PI.f64) 4)) (neg.f64 f))) (-.f64 (pow.f64 (exp.f64 (/.f64 (PI.f64) 4)) f) (pow.f64 (exp.f64 (/.f64 (PI.f64) 4)) (neg.f64 f))))) (/.f64 -4 (PI.f64))): 1 points increase in error, 0 points decrease in error
    (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) (neg.f64 f))))) (-.f64 (pow.f64 (exp.f64 (/.f64 (PI.f64) 4)) f) (pow.f64 (exp.f64 (/.f64 (PI.f64) 4)) (neg.f64 f))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
    (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f))))) (-.f64 (pow.f64 (exp.f64 (/.f64 (PI.f64) 4)) f) (pow.f64 (exp.f64 (/.f64 (PI.f64) 4)) (neg.f64 f))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
    (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f))) (pow.f64 (exp.f64 (/.f64 (PI.f64) 4)) (neg.f64 f))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 1 points decrease in error
    (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (Rewrite<= exp-prod_binary64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) (neg.f64 f))))))) (/.f64 -4 (PI.f64))): 1 points increase in error, 0 points decrease in error
    (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f))))))) (/.f64 -4 (PI.f64))): 0 points increase in error, 0 points decrease in error
    (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))))) (/.f64 (Rewrite<= metadata-eval (neg.f64 4)) (PI.f64))): 0 points increase in error, 0 points decrease in error
    (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))))) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 4 (PI.f64))))): 0 points increase in error, 0 points decrease in error
    (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))))) (neg.f64 (/.f64 (Rewrite<= metadata-eval (*.f64 1 4)) (PI.f64)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))))) (neg.f64 (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 1 (PI.f64)) 4)))): 0 points increase in error, 0 points decrease in error
    (*.f64 (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))))) (neg.f64 (Rewrite<= associate-/r/_binary64 (/.f64 1 (/.f64 (PI.f64) 4))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= *-commutative_binary64 (*.f64 (neg.f64 (/.f64 1 (/.f64 (PI.f64) 4))) (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= distribute-lft-neg-in_binary64 (neg.f64 (*.f64 (/.f64 1 (/.f64 (PI.f64) 4)) (log.f64 (/.f64 (+.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f)))) (-.f64 (exp.f64 (*.f64 (/.f64 (PI.f64) 4) f)) (exp.f64 (neg.f64 (*.f64 (/.f64 (PI.f64) 4) f))))))))): 0 points increase in error, 0 points decrease in error

  3. Taylor expanded in f around 0 2.3

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

  4. Simplified2.3

    \[\leadsto \log \left(\frac{{\left(e^{\frac{\pi}{4}}\right)}^{f} + {\left(e^{\frac{\pi}{4}}\right)}^{\left(-f\right)}}{\color{blue}{\mathsf{fma}\left({\pi}^{3}, {f}^{3} \cdot 0.005208333333333333, \mathsf{fma}\left(\pi, f \cdot 0.5, {\pi}^{5} \cdot \left({f}^{5} \cdot 1.6276041666666666 \cdot 10^{-5}\right)\right)\right)}}\right) \cdot \frac{-4}{\pi} \]
    Proof
    (fma.f64 (pow.f64 (PI.f64) 3) (*.f64 (pow.f64 f 3) 1/192) (fma.f64 (PI.f64) (*.f64 f 1/2) (*.f64 (pow.f64 (PI.f64) 5) (*.f64 (pow.f64 f 5) 1/61440)))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (pow.f64 (PI.f64) 3) (*.f64 (pow.f64 f 3) (Rewrite<= metadata-eval (-.f64 1/384 -1/384))) (fma.f64 (PI.f64) (*.f64 f 1/2) (*.f64 (pow.f64 (PI.f64) 5) (*.f64 (pow.f64 f 5) 1/61440)))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (pow.f64 (PI.f64) 3) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 1/384 -1/384) (pow.f64 f 3))) (fma.f64 (PI.f64) (*.f64 f 1/2) (*.f64 (pow.f64 (PI.f64) 5) (*.f64 (pow.f64 f 5) 1/61440)))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (pow.f64 (PI.f64) 3) (*.f64 (-.f64 1/384 -1/384) (pow.f64 f 3)) (fma.f64 (PI.f64) (*.f64 f (Rewrite<= metadata-eval (/.f64 1 2))) (*.f64 (pow.f64 (PI.f64) 5) (*.f64 (pow.f64 f 5) 1/61440)))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (pow.f64 (PI.f64) 3) (*.f64 (-.f64 1/384 -1/384) (pow.f64 f 3)) (fma.f64 (PI.f64) (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 1 2) f)) (*.f64 (pow.f64 (PI.f64) 5) (*.f64 (pow.f64 f 5) 1/61440)))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (pow.f64 (PI.f64) 3) (*.f64 (-.f64 1/384 -1/384) (pow.f64 f 3)) (fma.f64 (PI.f64) (*.f64 (/.f64 1 2) f) (*.f64 (pow.f64 (PI.f64) 5) (*.f64 (pow.f64 f 5) (Rewrite<= metadata-eval (-.f64 1/122880 -1/122880)))))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (pow.f64 (PI.f64) 3) (*.f64 (-.f64 1/384 -1/384) (pow.f64 f 3)) (fma.f64 (PI.f64) (*.f64 (/.f64 1 2) f) (*.f64 (pow.f64 (PI.f64) 5) (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 1/122880 -1/122880) (pow.f64 f 5)))))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (pow.f64 (PI.f64) 3) (*.f64 (-.f64 1/384 -1/384) (pow.f64 f 3)) (fma.f64 (PI.f64) (*.f64 (/.f64 1 2) f) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (PI.f64) 5) (-.f64 1/122880 -1/122880)) (pow.f64 f 5))))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (pow.f64 (PI.f64) 3) (*.f64 (-.f64 1/384 -1/384) (pow.f64 f 3)) (fma.f64 (PI.f64) (*.f64 (/.f64 1 2) f) (*.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 1/122880 (pow.f64 (PI.f64) 5)) (*.f64 -1/122880 (pow.f64 (PI.f64) 5)))) (pow.f64 f 5)))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (pow.f64 (PI.f64) 3) (*.f64 (-.f64 1/384 -1/384) (pow.f64 f 3)) (fma.f64 (PI.f64) (*.f64 (/.f64 1 2) f) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 f 5) (-.f64 (*.f64 1/122880 (pow.f64 (PI.f64) 5)) (*.f64 -1/122880 (pow.f64 (PI.f64) 5))))))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (pow.f64 (PI.f64) 3) (*.f64 (-.f64 1/384 -1/384) (pow.f64 f 3)) (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (PI.f64) (*.f64 (/.f64 1 2) f)) (*.f64 (pow.f64 f 5) (-.f64 (*.f64 1/122880 (pow.f64 (PI.f64) 5)) (*.f64 -1/122880 (pow.f64 (PI.f64) 5))))))): 1 points increase in error, 0 points decrease in error
    (fma.f64 (pow.f64 (PI.f64) 3) (*.f64 (-.f64 1/384 -1/384) (pow.f64 f 3)) (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (PI.f64) (/.f64 1 2)) f)) (*.f64 (pow.f64 f 5) (-.f64 (*.f64 1/122880 (pow.f64 (PI.f64) 5)) (*.f64 -1/122880 (pow.f64 (PI.f64) 5)))))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (pow.f64 (PI.f64) 3) (*.f64 (-.f64 1/384 -1/384) (pow.f64 f 3)) (+.f64 (*.f64 (*.f64 (PI.f64) (Rewrite=> metadata-eval 1/2)) f) (*.f64 (pow.f64 f 5) (-.f64 (*.f64 1/122880 (pow.f64 (PI.f64) 5)) (*.f64 -1/122880 (pow.f64 (PI.f64) 5)))))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (pow.f64 (PI.f64) 3) (*.f64 (-.f64 1/384 -1/384) (pow.f64 f 3)) (+.f64 (*.f64 (*.f64 (PI.f64) (Rewrite<= metadata-eval (-.f64 1/4 -1/4))) f) (*.f64 (pow.f64 f 5) (-.f64 (*.f64 1/122880 (pow.f64 (PI.f64) 5)) (*.f64 -1/122880 (pow.f64 (PI.f64) 5)))))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (pow.f64 (PI.f64) 3) (*.f64 (-.f64 1/384 -1/384) (pow.f64 f 3)) (+.f64 (*.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 1/4 (PI.f64)) (*.f64 -1/4 (PI.f64)))) f) (*.f64 (pow.f64 f 5) (-.f64 (*.f64 1/122880 (pow.f64 (PI.f64) 5)) (*.f64 -1/122880 (pow.f64 (PI.f64) 5)))))): 0 points increase in error, 0 points decrease in error
    (fma.f64 (pow.f64 (PI.f64) 3) (*.f64 (-.f64 1/384 -1/384) (pow.f64 f 3)) (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 (pow.f64 f 5) (-.f64 (*.f64 1/122880 (pow.f64 (PI.f64) 5)) (*.f64 -1/122880 (pow.f64 (PI.f64) 5)))) (*.f64 (-.f64 (*.f64 1/4 (PI.f64)) (*.f64 -1/4 (PI.f64))) f)))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (pow.f64 (PI.f64) 3) (*.f64 (-.f64 1/384 -1/384) (pow.f64 f 3))) (+.f64 (*.f64 (pow.f64 f 5) (-.f64 (*.f64 1/122880 (pow.f64 (PI.f64) 5)) (*.f64 -1/122880 (pow.f64 (PI.f64) 5)))) (*.f64 (-.f64 (*.f64 1/4 (PI.f64)) (*.f64 -1/4 (PI.f64))) f)))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (pow.f64 (PI.f64) 3) (-.f64 1/384 -1/384)) (pow.f64 f 3))) (+.f64 (*.f64 (pow.f64 f 5) (-.f64 (*.f64 1/122880 (pow.f64 (PI.f64) 5)) (*.f64 -1/122880 (pow.f64 (PI.f64) 5)))) (*.f64 (-.f64 (*.f64 1/4 (PI.f64)) (*.f64 -1/4 (PI.f64))) f))): 0 points increase in error, 0 points decrease in error
    (+.f64 (*.f64 (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 1/384 (pow.f64 (PI.f64) 3)) (*.f64 -1/384 (pow.f64 (PI.f64) 3)))) (pow.f64 f 3)) (+.f64 (*.f64 (pow.f64 f 5) (-.f64 (*.f64 1/122880 (pow.f64 (PI.f64) 5)) (*.f64 -1/122880 (pow.f64 (PI.f64) 5)))) (*.f64 (-.f64 (*.f64 1/4 (PI.f64)) (*.f64 -1/4 (PI.f64))) f))): 0 points increase in error, 0 points decrease in error
    (+.f64 (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 f 3) (-.f64 (*.f64 1/384 (pow.f64 (PI.f64) 3)) (*.f64 -1/384 (pow.f64 (PI.f64) 3))))) (+.f64 (*.f64 (pow.f64 f 5) (-.f64 (*.f64 1/122880 (pow.f64 (PI.f64) 5)) (*.f64 -1/122880 (pow.f64 (PI.f64) 5)))) (*.f64 (-.f64 (*.f64 1/4 (PI.f64)) (*.f64 -1/4 (PI.f64))) f))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 (pow.f64 f 5) (-.f64 (*.f64 1/122880 (pow.f64 (PI.f64) 5)) (*.f64 -1/122880 (pow.f64 (PI.f64) 5)))) (*.f64 (-.f64 (*.f64 1/4 (PI.f64)) (*.f64 -1/4 (PI.f64))) f)) (*.f64 (pow.f64 f 3) (-.f64 (*.f64 1/384 (pow.f64 (PI.f64) 3)) (*.f64 -1/384 (pow.f64 (PI.f64) 3)))))): 0 points increase in error, 0 points decrease in error
    (Rewrite<= associate-+r+_binary64 (+.f64 (*.f64 (pow.f64 f 5) (-.f64 (*.f64 1/122880 (pow.f64 (PI.f64) 5)) (*.f64 -1/122880 (pow.f64 (PI.f64) 5)))) (+.f64 (*.f64 (-.f64 (*.f64 1/4 (PI.f64)) (*.f64 -1/4 (PI.f64))) f) (*.f64 (pow.f64 f 3) (-.f64 (*.f64 1/384 (pow.f64 (PI.f64) 3)) (*.f64 -1/384 (pow.f64 (PI.f64) 3))))))): 1 points increase in error, 0 points decrease in error

  5. Final simplification2.3

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

Alternatives

Alternative 1
Error2.4
Cost59136
\[\left(f \cdot f\right) \cdot \left(\pi \cdot -0.08333333333333333\right) + \mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{f} \cdot \frac{1}{\pi}\right)}{\pi}, {f}^{4} \cdot \left(\left(\pi \cdot \left({\pi}^{2} \cdot 0.006944444444444444\right)\right) \cdot 0.125\right)\right) \]
Alternative 2
Error2.3
Cost59008
\[\left(f \cdot f\right) \cdot \left(\pi \cdot -0.08333333333333333\right) + \mathsf{fma}\left(-4, \frac{\log \left(\frac{4}{\pi \cdot f}\right)}{\pi}, {f}^{4} \cdot \left(\left(\pi \cdot \left({\pi}^{2} \cdot 0.006944444444444444\right)\right) \cdot 0.125\right)\right) \]
Alternative 3
Error2.3
Cost45760
\[\frac{-4 \cdot \log \left(\mathsf{fma}\left(\mathsf{fma}\left(\pi \cdot 0.010416666666666666, -4, \pi \cdot 0.125\right), f, \frac{4}{\pi} \cdot \frac{1}{f}\right)\right)}{\pi} \]
Alternative 4
Error2.4
Cost32640
\[\frac{-4}{\pi} \cdot \log \left(\mathsf{fma}\left(f, \pi \cdot 0.08333333333333333, \frac{\frac{4}{\pi}}{f}\right)\right) \]
Alternative 5
Error2.3
Cost32640
\[\frac{-4 \cdot \log \left(\mathsf{fma}\left(\pi \cdot 0.08333333333333333, f, \frac{4}{\pi \cdot f}\right)\right)}{\pi} \]
Alternative 6
Error2.6
Cost26048
\[\log \left(\sqrt[3]{\frac{4}{\pi \cdot f}}\right) \cdot \frac{-12}{\pi} \]
Alternative 7
Error2.8
Cost19648
\[\frac{-4}{\pi} \cdot \log \left(\frac{\frac{4}{\pi}}{f}\right) \]
Alternative 8
Error2.8
Cost19648
\[\frac{-4}{\frac{\pi}{\log \left(\frac{4}{\pi \cdot f}\right)}} \]
Alternative 9
Error2.7
Cost19648
\[\frac{-4 \cdot \log \left(\frac{4}{\pi \cdot f}\right)}{\pi} \]

Error

Reproduce

herbie shell --seed 2022325 
(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)))))))))