Average Error: 0.4 → 0.1
Time: 19.7s
Precision: binary64
Cost: 7104
\[\frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120 \]
↓
\[\mathsf{fma}\left(a, 120, \frac{60}{z - t} \cdot \left(x - y\right)\right) \]
(FPCore (x y z t a)
 :precision binary64
 (+ (/ (* 60.0 (- x y)) (- z t)) (* a 120.0)))
↓
(FPCore (x y z t a)
 :precision binary64
 (fma a 120.0 (* (/ 60.0 (- z t)) (- x y))))
double code(double x, double y, double z, double t, double a) {
	return ((60.0 * (x - y)) / (z - t)) + (a * 120.0);
}
↓
double code(double x, double y, double z, double t, double a) {
	return fma(a, 120.0, ((60.0 / (z - t)) * (x - y)));
}

function code(x, y, z, t, a)
	return Float64(Float64(Float64(60.0 * Float64(x - y)) / Float64(z - t)) + Float64(a * 120.0))
end
↓
function code(x, y, z, t, a)
	return fma(a, 120.0, Float64(Float64(60.0 / Float64(z - t)) * Float64(x - y)))
end

code[x_, y_, z_, t_, a_] := N[(N[(N[(60.0 * N[(x - y), $MachinePrecision]), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision] + N[(a * 120.0), $MachinePrecision]), $MachinePrecision]
↓
code[x_, y_, z_, t_, a_] := N[(a * 120.0 + N[(N[(60.0 / N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

\frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120
↓
\mathsf{fma}\left(a, 120, \frac{60}{z - t} \cdot \left(x - y\right)\right)

Error

Target

Original0.4
Target0.2
Herbie0.1
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]

Derivation

  1. Initial program 0.4

    \[\frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120 \]

  2. Simplified0.1

    \[\leadsto \color{blue}{\mathsf{fma}\left(a, 120, \frac{60}{z - t} \cdot \left(x - y\right)\right)} \]
    Proof

    [Start]0.4

    \[ \frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120 \]

    +-commutative [=>]0.4

    \[ \color{blue}{a \cdot 120 + \frac{60 \cdot \left(x - y\right)}{z - t}} \]

    fma-def [=>]0.4

    \[ \color{blue}{\mathsf{fma}\left(a, 120, \frac{60 \cdot \left(x - y\right)}{z - t}\right)} \]

    associate-*l/ [<=]0.1

    \[ \mathsf{fma}\left(a, 120, \color{blue}{\frac{60}{z - t} \cdot \left(x - y\right)}\right) \]

  3. Final simplification0.1

    \[\leadsto \mathsf{fma}\left(a, 120, \frac{60}{z - t} \cdot \left(x - y\right)\right) \]

Alternatives

Alternative 1
Error15.9
Cost2656
\[\begin{array}{l} t_1 := 60 \cdot \frac{x - y}{z - t}\\ t_2 := a \cdot 120 + 60 \cdot \frac{y}{t}\\ t_3 := \frac{60}{z - t} \cdot \left(x - y\right)\\ \mathbf{if}\;a \cdot 120 \leq -1 \cdot 10^{+44}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq -2 \cdot 10^{-37}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;a \cdot 120 \leq -5 \cdot 10^{-80}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot 120 \leq 5 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 120 \leq 2 \cdot 10^{-46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \cdot 120 \leq 1:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 120 \leq 2 \cdot 10^{+49}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq 10^{+77}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 2
Error16.2
Cost2136
\[\begin{array}{l} t_1 := \frac{60}{z - t} \cdot \left(x - y\right)\\ \mathbf{if}\;a \cdot 120 \leq -1 \cdot 10^{+44}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq -2 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 120 \leq -5 \cdot 10^{-80}:\\ \;\;\;\;a \cdot 120 + 60 \cdot \frac{y}{t}\\ \mathbf{elif}\;a \cdot 120 \leq 10^{-77}:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \mathbf{elif}\;a \cdot 120 \leq 2 \cdot 10^{+49}:\\ \;\;\;\;a \cdot 120 + -60 \cdot \frac{y}{z}\\ \mathbf{elif}\;a \cdot 120 \leq 10^{+77}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 3
Error15.6
Cost1364
\[\begin{array}{l} t_1 := a \cdot 120 + \left(x - y\right) \cdot \frac{60}{z}\\ t_2 := 60 \cdot \frac{x - y}{z - t}\\ \mathbf{if}\;z \leq -2.8 \cdot 10^{+21}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -5.5 \cdot 10^{-43}:\\ \;\;\;\;a \cdot 120 + \frac{-60}{\frac{t}{x}}\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-144}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-44}:\\ \;\;\;\;a \cdot 120 + 60 \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+24}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error13.8
Cost1364
\[\begin{array}{l} t_1 := a \cdot 120 + \left(x - y\right) \cdot \frac{60}{z}\\ t_2 := a \cdot 120 + \frac{y \cdot -60}{z - t}\\ \mathbf{if}\;z \leq -1.3 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.25 \cdot 10^{-90}:\\ \;\;\;\;a \cdot 120 + \frac{-60}{\frac{t}{x}}\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-27}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 0.00095:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \mathbf{elif}\;z \leq 10^{+61}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error10.6
Cost1232
\[\begin{array}{l} t_1 := a \cdot 120 + \frac{x - y}{\frac{t}{-60}}\\ t_2 := a \cdot 120 + \left(x - y\right) \cdot \frac{60}{z}\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+19}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.7 \cdot 10^{-90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -6.4 \cdot 10^{-144}:\\ \;\;\;\;a \cdot 120 + \frac{y \cdot -60}{z - t}\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+59}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 6
Error15.6
Cost1105
\[\begin{array}{l} \mathbf{if}\;a \leq -7.5 \cdot 10^{+27}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1050000 \lor \neg \left(a \leq 1.82 \cdot 10^{+50}\right) \land a \leq 1.05 \cdot 10^{+75}:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 7
Error15.6
Cost1104
\[\begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{+29}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 14800:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{+49}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{+74}:\\ \;\;\;\;\frac{60}{z - t} \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 8
Error26.8
Cost978
\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+152} \lor \neg \left(y \leq 3 \cdot 10^{+186}\right) \land \left(y \leq 1.8 \cdot 10^{+222} \lor \neg \left(y \leq 8.6 \cdot 10^{+252}\right)\right):\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 9
Error29.8
Cost848
\[\begin{array}{l} t_1 := -60 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -7 \cdot 10^{+156}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+208}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+220}:\\ \;\;\;\;60 \cdot \frac{y}{t}\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+255}:\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 10
Error0.2
Cost832
\[a \cdot 120 + \frac{60}{z - t} \cdot \left(x - y\right) \]
Alternative 11
Error29.8
Cost585
\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+157} \lor \neg \left(y \leq 3.95 \cdot 10^{+257}\right):\\ \;\;\;\;-60 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 12
Error29.3
Cost584
\[\begin{array}{l} \mathbf{if}\;a \leq -7.6 \cdot 10^{-108}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-228}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 13
Error29.1
Cost192
\[a \cdot 120 \]

Error

Reproduce

herbie shell --seed 2022354 
(FPCore (x y z t a)
  :name "Data.Colour.RGB:hslsv from colour-2.3.3, B"
  :precision binary64

  :herbie-target
  (+ (/ 60.0 (/ (- z t) (- x y))) (* a 120.0))

  (+ (/ (* 60.0 (- x y)) (- z t)) (* a 120.0)))