Average Error: 20.7 → 17.2
Time: 19.5s
Precision: binary64
Cost: 19776
\[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]
↓
\[\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{\frac{a}{-3}}{b}\right) \]
(FPCore (x y z t a b)
 :precision binary64
 (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))
↓
(FPCore (x y z t a b)
 :precision binary64
 (fma 2.0 (* (sqrt x) (cos y)) (/ (/ a -3.0) b)))
double code(double x, double y, double z, double t, double a, double b) {
	return ((2.0 * sqrt(x)) * cos((y - ((z * t) / 3.0)))) - (a / (b * 3.0));
}
↓
double code(double x, double y, double z, double t, double a, double b) {
	return fma(2.0, (sqrt(x) * cos(y)), ((a / -3.0) / b));
}

function code(x, y, z, t, a, b)
	return Float64(Float64(Float64(2.0 * sqrt(x)) * cos(Float64(y - Float64(Float64(z * t) / 3.0)))) - Float64(a / Float64(b * 3.0)))
end
↓
function code(x, y, z, t, a, b)
	return fma(2.0, Float64(sqrt(x) * cos(y)), Float64(Float64(a / -3.0) / b))
end

code[x_, y_, z_, t_, a_, b_] := N[(N[(N[(2.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(y - N[(N[(z * t), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - N[(a / N[(b * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_, b_] := N[(2.0 * N[(N[Sqrt[x], $MachinePrecision] * N[Cos[y], $MachinePrecision]), $MachinePrecision] + N[(N[(a / -3.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3}
↓
\mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{\frac{a}{-3}}{b}\right)

Error

Target

Original20.7
Target19.0
Herbie17.2
\[\begin{array}{l} \mathbf{if}\;z < -1.3793337487235141 \cdot 10^{+129}:\\ \;\;\;\;\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(\frac{1}{y} - \frac{\frac{0.3333333333333333}{z}}{t}\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{elif}\;z < 3.516290613555987 \cdot 10^{+106}:\\ \;\;\;\;\left(\sqrt{x} \cdot 2\right) \cdot \cos \left(y - \frac{t}{3} \cdot z\right) - \frac{\frac{a}{3}}{b}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(y - \frac{\frac{0.3333333333333333}{z}}{t}\right) \cdot \left(2 \cdot \sqrt{x}\right) - \frac{\frac{a}{b}}{3}\\ \end{array} \]

Derivation

  1. Initial program 20.7

    \[\left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

  2. Simplified20.6

    \[\leadsto \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(\mathsf{fma}\left(z, t \cdot -0.3333333333333333, y\right)\right), \frac{\frac{a}{-3}}{b}\right)} \]
    Proof

    [Start]20.7

    \[ \left(2 \cdot \sqrt{x}\right) \cdot \cos \left(y - \frac{z \cdot t}{3}\right) - \frac{a}{b \cdot 3} \]

    associate-*l* [=>]20.7

    \[ \color{blue}{2 \cdot \left(\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)} - \frac{a}{b \cdot 3} \]

    fma-neg [=>]20.7

    \[ \color{blue}{\mathsf{fma}\left(2, \sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right), -\frac{a}{b \cdot 3}\right)} \]

    remove-double-neg [<=]20.7

    \[ \mathsf{fma}\left(2, \color{blue}{-\left(-\sqrt{x} \cdot \cos \left(y - \frac{z \cdot t}{3}\right)\right)}, -\frac{a}{b \cdot 3}\right) \]

  3. Taylor expanded in z around 0 17.2

    \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \color{blue}{\cos y}, \frac{\frac{a}{-3}}{b}\right) \]

  4. Final simplification17.2

    \[\leadsto \mathsf{fma}\left(2, \sqrt{x} \cdot \cos y, \frac{\frac{a}{-3}}{b}\right) \]

Alternatives

Alternative 1
Error21.7
Cost14024
\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t_1 \leq -50000000:\\ \;\;\;\;2 \cdot \sqrt{x} - t_1\\ \mathbf{elif}\;t_1 \leq 10^{-138}:\\ \;\;\;\;\sqrt{x} \cdot \left(2 \cdot \cos y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \sqrt{x}, \frac{\frac{a}{-3}}{b}\right)\\ \end{array} \]
Alternative 2
Error21.7
Cost13897
\[\begin{array}{l} t_1 := \frac{a}{b \cdot 3}\\ \mathbf{if}\;t_1 \leq -50000000 \lor \neg \left(t_1 \leq 10^{-138}\right):\\ \;\;\;\;2 \cdot \sqrt{x} - t_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x} \cdot \left(2 \cdot \cos y\right)\\ \end{array} \]
Alternative 3
Error17.3
Cost13504
\[\frac{-0.3333333333333333}{\frac{b}{a}} + 2 \cdot \left(\sqrt{x} \cdot \cos y\right) \]
Alternative 4
Error17.3
Cost13504
\[\cos y \cdot \left(2 \cdot \sqrt{x}\right) - \frac{a}{b \cdot 3} \]
Alternative 5
Error25.3
Cost6976
\[2 \cdot \sqrt{x} - \frac{a}{b \cdot 3} \]
Alternative 6
Error36.1
Cost320
\[-0.3333333333333333 \cdot \frac{a}{b} \]
Alternative 7
Error36.1
Cost320
\[\frac{-0.3333333333333333}{\frac{b}{a}} \]
Alternative 8
Error36.0
Cost320
\[\frac{a}{\frac{b}{-0.3333333333333333}} \]
Alternative 9
Error36.0
Cost320
\[\frac{\frac{a}{-3}}{b} \]

Error

Reproduce

herbie shell --seed 2023016 
(FPCore (x y z t a b)
  :name "Diagrams.Solve.Polynomial:cubForm  from diagrams-solve-0.1, K"
  :precision binary64

  :herbie-target
  (if (< z -1.3793337487235141e+129) (- (* (* 2.0 (sqrt x)) (cos (- (/ 1.0 y) (/ (/ 0.3333333333333333 z) t)))) (/ (/ a 3.0) b)) (if (< z 3.516290613555987e+106) (- (* (* (sqrt x) 2.0) (cos (- y (* (/ t 3.0) z)))) (/ (/ a 3.0) b)) (- (* (cos (- y (/ (/ 0.3333333333333333 z) t))) (* 2.0 (sqrt x))) (/ (/ a b) 3.0))))

  (- (* (* 2.0 (sqrt x)) (cos (- y (/ (* z t) 3.0)))) (/ a (* b 3.0))))