?

Average Error: 0.69% → 0.2%
Time: 20.6s
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

Original0.69%
Target0.25%
Herbie0.2%
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]

Derivation?

  1. Initial program 0.69

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

  2. Simplified0.2

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

    [Start]0.69

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

    +-commutative [=>]0.69

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

    fma-def [=>]0.66

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

    associate-*l/ [<=]0.2

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

  3. Final simplification0.2

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

Alternatives

Alternative 1
Error18.85%
Cost1746
\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -2 \cdot 10^{-38} \lor \neg \left(a \cdot 120 \leq 10^{-173} \lor \neg \left(a \cdot 120 \leq 10^{-75}\right) \land a \cdot 120 \leq 1.2 \cdot 10^{+29}\right):\\ \;\;\;\;\frac{y \cdot -60}{z - t} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \end{array} \]
Alternative 2
Error47.3%
Cost1376
\[\begin{array}{l} t_1 := -60 \cdot \frac{y}{z}\\ t_2 := 60 \cdot \frac{y}{t}\\ \mathbf{if}\;a \leq -2.1 \cdot 10^{-43}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -4 \cdot 10^{-97}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -1.6 \cdot 10^{-107}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq -2.4 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-239}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-186}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-31}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 3.4 \cdot 10^{+23}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 3
Error47.28%
Cost1376
\[\begin{array}{l} t_1 := -60 \cdot \frac{y}{z}\\ t_2 := 60 \cdot \frac{y}{t}\\ \mathbf{if}\;a \leq -2.1 \cdot 10^{-43}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -3.2 \cdot 10^{-98}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -8 \cdot 10^{-108}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq -9.5 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-241}:\\ \;\;\;\;\frac{y}{\frac{t}{60}}\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-183}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+20}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 4
Error25.07%
Cost1232
\[\begin{array}{l} t_1 := \frac{60}{\frac{z}{x - y}} + a \cdot 120\\ \mathbf{if}\;z \leq -1.9 \cdot 10^{+35}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.4 \cdot 10^{-238}:\\ \;\;\;\;\frac{y}{\frac{t}{60}} + a \cdot 120\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-174}:\\ \;\;\;\;a \cdot 120 + -60 \cdot \frac{x}{t}\\ \mathbf{elif}\;z \leq 6.9 \cdot 10^{-37}:\\ \;\;\;\;\frac{60}{z - t} \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error40.45%
Cost1108
\[\begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-41}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-180}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-78}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 4.7 \cdot 10^{-78}:\\ \;\;\;\;\frac{-60}{\frac{z}{y}}\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{+20}:\\ \;\;\;\;-60 \cdot \frac{x - y}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 6
Error40.59%
Cost1108
\[\begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{-42}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-183}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-78}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-71}:\\ \;\;\;\;60 \cdot \frac{x - y}{z}\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{+20}:\\ \;\;\;\;-60 \cdot \frac{x - y}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 7
Error40.44%
Cost1108
\[\begin{array}{l} \mathbf{if}\;a \leq -5.3 \cdot 10^{-43}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-180}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-78}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{-72}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{60}{z}\\ \mathbf{elif}\;a \leq 9 \cdot 10^{+20}:\\ \;\;\;\;-60 \cdot \frac{x - y}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 8
Error23.4%
Cost1096
\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -1000000:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq 4 \cdot 10^{+60}:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 9
Error40.3%
Cost977
\[\begin{array}{l} \mathbf{if}\;a \leq -2.15 \cdot 10^{-41}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-180} \lor \neg \left(a \leq 6.5 \cdot 10^{-31}\right) \land a \leq 10^{+27}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 10
Error10.88%
Cost969
\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+56} \lor \neg \left(y \leq 9 \cdot 10^{+53}\right):\\ \;\;\;\;\frac{y \cdot -60}{z - t} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;\frac{60 \cdot x}{z - t} + a \cdot 120\\ \end{array} \]
Alternative 11
Error0.25%
Cost832
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]
Alternative 12
Error45.16%
Cost584
\[\begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{-82}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{-184}:\\ \;\;\;\;-60 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 13
Error45.96%
Cost192
\[a \cdot 120 \]

Error

Reproduce?

herbie shell --seed 2023089 
(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)))