Average Error: 14.8 → 0.3
Time: 4.1s
Precision: binary64
\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]
↓
\[\frac{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b + a}}{\left(b - a\right) \cdot 2} \]
(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))
↓
(FPCore (a b)
 :precision binary64
 (/ (/ (fma PI (/ -1.0 b) (/ PI a)) (+ b a)) (* (- b a) 2.0)))
double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}
↓
double code(double a, double b) {
	return (fma(((double) M_PI), (-1.0 / b), (((double) M_PI) / a)) / (b + a)) / ((b - a) * 2.0);
}

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end
↓
function code(a, b)
	return Float64(Float64(fma(pi, Float64(-1.0 / b), Float64(pi / a)) / Float64(b + a)) / Float64(Float64(b - a) * 2.0))
end

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[a_, b_] := N[(N[(N[(Pi * N[(-1.0 / b), $MachinePrecision] + N[(Pi / a), $MachinePrecision]), $MachinePrecision] / N[(b + a), $MachinePrecision]), $MachinePrecision] / N[(N[(b - a), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right)
↓
\frac{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b + a}}{\left(b - a\right) \cdot 2}

Error

Derivation

  1. Initial program 14.8

    \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

  2. Simplified14.8

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{2}}{b \cdot b - a \cdot a}} \]

  3. Applied egg-rr0.3

    \[\leadsto \color{blue}{\left(\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b + a} \cdot \frac{0.5}{b - a}\right) \cdot 1} \]

  4. Applied egg-rr0.3

    \[\leadsto \left(\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \color{blue}{{\left(\frac{a}{\pi}\right)}^{-1}}\right)}{b + a} \cdot \frac{0.5}{b - a}\right) \cdot 1 \]

  5. Applied egg-rr0.3

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{a + b}}{\left(b - a\right) \cdot 2}} \cdot 1 \]

  6. Final simplification0.3

    \[\leadsto \frac{\frac{\mathsf{fma}\left(\pi, \frac{-1}{b}, \frac{\pi}{a}\right)}{b + a}}{\left(b - a\right) \cdot 2} \]

Reproduce

herbie shell --seed 2022190 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))