?

Average Error: 0.6% → 0.2%
Time: 22.2s
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

Original0.6%
Target0.24%
Herbie0.2%
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]

Derivation?

  1. Initial program 0.6

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

  2. Simplified0.2

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

    [Start]0.6

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

    +-commutative [=>]0.6

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

    fma-def [=>]0.57

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

    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
Error0.22%
Cost7104
\[\mathsf{fma}\left(60, \frac{x - y}{z - t}, a \cdot 120\right) \]
Alternative 2
Error27.87%
Cost2004
\[\begin{array}{l} t_1 := a \cdot 120 + \frac{60}{\frac{z}{x - y}}\\ \mathbf{if}\;a \cdot 120 \leq -4 \cdot 10^{+111}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq -5 \cdot 10^{+62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \cdot 120 \leq -1 \cdot 10^{-49}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq 10^{-122}:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \mathbf{elif}\;a \cdot 120 \leq 3 \cdot 10^{+33}:\\ \;\;\;\;a \cdot 120 + \frac{x}{\frac{t}{-60}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 3
Error27.06%
Cost1356
\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -1 \cdot 10^{-49}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq 10^{-122}:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \mathbf{elif}\;a \cdot 120 \leq 3 \cdot 10^{+33}:\\ \;\;\;\;a \cdot 120 + \frac{x}{\frac{t}{-60}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120 + 60 \cdot \frac{x}{z}\\ \end{array} \]
Alternative 4
Error44.41%
Cost1244
\[\begin{array}{l} t_1 := 60 \cdot \frac{x}{z}\\ \mathbf{if}\;a \leq -1 \cdot 10^{-137}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -2.35 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-217}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -9.2 \cdot 10^{-306}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-234}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-204}:\\ \;\;\;\;60 \cdot \frac{y}{t}\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 5
Error44.44%
Cost1244
\[\begin{array}{l} t_1 := 60 \cdot \frac{x}{z}\\ \mathbf{if}\;a \leq -1.35 \cdot 10^{-137}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -2.45 \cdot 10^{-197}:\\ \;\;\;\;\frac{x}{z \cdot 0.016666666666666666}\\ \mathbf{elif}\;a \leq -8.5 \cdot 10^{-218}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -4.5 \cdot 10^{-305}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-234}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-204}:\\ \;\;\;\;60 \cdot \frac{y}{t}\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 6
Error44.37%
Cost1244
\[\begin{array}{l} t_1 := 60 \cdot \frac{x}{z}\\ \mathbf{if}\;a \leq -4.4 \cdot 10^{-140}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -2.7 \cdot 10^{-197}:\\ \;\;\;\;\frac{x}{z \cdot 0.016666666666666666}\\ \mathbf{elif}\;a \leq -4.9 \cdot 10^{-216}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -3.9 \cdot 10^{-305}:\\ \;\;\;\;\frac{x \cdot -60}{t}\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-234}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-203}:\\ \;\;\;\;60 \cdot \frac{y}{t}\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 7
Error40.45%
Cost1241
\[\begin{array}{l} t_1 := 60 \cdot \frac{x - y}{z}\\ \mathbf{if}\;a \leq -1.2 \cdot 10^{-79}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-204}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{-60}{t}\\ \mathbf{elif}\;a \leq 4.1 \cdot 10^{-122} \lor \neg \left(a \leq 2.8 \cdot 10^{+31}\right) \land a \leq 2.6 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 8
Error40.49%
Cost1240
\[\begin{array}{l} t_1 := 60 \cdot \frac{x - y}{z}\\ \mathbf{if}\;a \leq -5 \cdot 10^{-76}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -3.05 \cdot 10^{-212}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-204}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{-60}{t}\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-123}:\\ \;\;\;\;\frac{60}{\frac{z}{x - y}}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+31}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 9
Error40.56%
Cost1240
\[\begin{array}{l} t_1 := 60 \cdot \frac{x - y}{z}\\ \mathbf{if}\;a \leq -1.7 \cdot 10^{-71}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-215}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-204}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{-60}{t}\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{-120}:\\ \;\;\;\;\frac{x - y}{\frac{z}{60}}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+31}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 10
Error16.4%
Cost1225
\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -1 \cdot 10^{-49} \lor \neg \left(a \cdot 120 \leq 10^{-122}\right):\\ \;\;\;\;\frac{y \cdot -60}{z - t} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \end{array} \]
Alternative 11
Error39.02%
Cost1108
\[\begin{array}{l} t_1 := -60 \cdot \frac{y}{z - t}\\ t_2 := 60 \cdot \frac{x}{z - t}\\ \mathbf{if}\;a \leq -3.9 \cdot 10^{-54}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -1.4 \cdot 10^{-243}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -5.8 \cdot 10^{-286}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{-217}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 12
Error39.91%
Cost1108
\[\begin{array}{l} t_1 := 60 \cdot \frac{x - y}{z}\\ \mathbf{if}\;a \leq -1.6 \cdot 10^{-71}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -1.4 \cdot 10^{-221}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-120}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{+31}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 13
Error26.01%
Cost1106
\[\begin{array}{l} \mathbf{if}\;a \leq -4.3 \cdot 10^{-53} \lor \neg \left(a \leq 3.1 \cdot 10^{-119}\right) \land \left(a \leq 2.8 \cdot 10^{+31} \lor \neg \left(a \leq 2.6 \cdot 10^{+65}\right)\right):\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \end{array} \]
Alternative 14
Error24.74%
Cost1104
\[\begin{array}{l} t_1 := 60 \cdot \frac{x - y}{z - t}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{-52}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 5.9 \cdot 10^{-53}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.05 \cdot 10^{+31}:\\ \;\;\;\;a \cdot 120 + 60 \cdot \frac{y}{t}\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 15
Error44.66%
Cost980
\[\begin{array}{l} t_1 := 60 \cdot \frac{x}{z}\\ \mathbf{if}\;a \leq -4 \cdot 10^{-137}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -2.2 \cdot 10^{-197}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.6 \cdot 10^{-216}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 5.5 \cdot 10^{-202}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-120}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 16
Error10.61%
Cost969
\[\begin{array}{l} \mathbf{if}\;x \leq -1.75 \cdot 10^{-66} \lor \neg \left(x \leq 1000000000\right):\\ \;\;\;\;\frac{60 \cdot x}{z - t} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -60}{z - t} + a \cdot 120\\ \end{array} \]
Alternative 17
Error0.24%
Cost832
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]
Alternative 18
Error38.72%
Cost712
\[\begin{array}{l} \mathbf{if}\;a \leq -1.02 \cdot 10^{-135}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 9.2 \cdot 10^{-125}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 19
Error44.01%
Cost584
\[\begin{array}{l} \mathbf{if}\;a \leq -1.8 \cdot 10^{-159}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-201}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 20
Error44.1%
Cost584
\[\begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{-136}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-141}:\\ \;\;\;\;-60 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 21
Error45.35%
Cost192
\[a \cdot 120 \]

Error

Reproduce?

herbie shell --seed 2023115 
(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)))