Average Error: 0.4 → 0.1
Time: 19.4s
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
Error10.6
Cost1225
\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -1 \cdot 10^{-41} \lor \neg \left(a \cdot 120 \leq 5 \cdot 10^{-79}\right):\\ \;\;\;\;\frac{y \cdot -60}{z - t} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;\frac{60 \cdot \left(x - y\right)}{z - t}\\ \end{array} \]
Alternative 2
Error28.2
Cost1112
\[\begin{array}{l} t_1 := -60 \cdot \frac{x}{t}\\ t_2 := -60 \cdot \frac{y}{z}\\ \mathbf{if}\;a \leq -2.4 \cdot 10^{-132}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -1.05 \cdot 10^{-200}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -5.4 \cdot 10^{-216}:\\ \;\;\;\;60 \cdot \frac{y}{t}\\ \mathbf{elif}\;a \leq -6.2 \cdot 10^{-239}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-306}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 8.6 \cdot 10^{-121}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 3
Error28.3
Cost1112
\[\begin{array}{l} t_1 := x \cdot \frac{-60}{t}\\ t_2 := -60 \cdot \frac{y}{z}\\ \mathbf{if}\;a \leq -5 \cdot 10^{-132}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -1 \cdot 10^{-201}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq -4.2 \cdot 10^{-214}:\\ \;\;\;\;60 \cdot \frac{y}{t}\\ \mathbf{elif}\;a \leq -6.1 \cdot 10^{-239}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-307}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 4
Error28.3
Cost1112
\[\begin{array}{l} t_1 := -60 \cdot \frac{y}{z}\\ \mathbf{if}\;a \leq -4.2 \cdot 10^{-132}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -7.2 \cdot 10^{-202}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -3.7 \cdot 10^{-216}:\\ \;\;\;\;60 \cdot \frac{y}{t}\\ \mathbf{elif}\;a \leq -1.8 \cdot 10^{-239}:\\ \;\;\;\;x \cdot \frac{-60}{t}\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-308}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{x \cdot -60}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 5
Error24.7
Cost1108
\[\begin{array}{l} t_1 := -60 \cdot \frac{y}{z - t}\\ \mathbf{if}\;a \leq -4.4 \cdot 10^{-44}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -6 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -2.8 \cdot 10^{-239}:\\ \;\;\;\;60 \cdot \frac{x}{z - t}\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-299}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9 \cdot 10^{-77}:\\ \;\;\;\;-60 \cdot \frac{x - y}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 6
Error24.7
Cost1108
\[\begin{array}{l} t_1 := -60 \cdot \frac{y}{z - t}\\ \mathbf{if}\;a \leq -3.8 \cdot 10^{-44}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -1.3 \cdot 10^{-211}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -9.6 \cdot 10^{-239}:\\ \;\;\;\;\frac{60}{z - t} \cdot x\\ \mathbf{elif}\;a \leq 8 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.1 \cdot 10^{-77}:\\ \;\;\;\;-60 \cdot \frac{x - y}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 7
Error24.7
Cost1108
\[\begin{array}{l} t_1 := -60 \cdot \frac{y}{z - t}\\ \mathbf{if}\;a \leq -4.4 \cdot 10^{-44}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -1.1 \cdot 10^{-208}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq -4.8 \cdot 10^{-239}:\\ \;\;\;\;\frac{60}{z - t} \cdot x\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-300}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{x - y}{\frac{t}{-60}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 8
Error14.9
Cost1096
\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -1 \cdot 10^{-41}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq 5 \cdot 10^{-67}:\\ \;\;\;\;\frac{60}{z - t} \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 9
Error15.0
Cost1096
\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -1 \cdot 10^{-41}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq 5 \cdot 10^{-67}:\\ \;\;\;\;\frac{60 \cdot \left(x - y\right)}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 10
Error11.8
Cost969
\[\begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{-126} \lor \neg \left(z \leq 2.7 \cdot 10^{-50}\right):\\ \;\;\;\;\frac{y \cdot -60}{z - t} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\frac{t}{-60}} + a \cdot 120\\ \end{array} \]
Alternative 11
Error11.1
Cost968
\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-126}:\\ \;\;\;\;\frac{y \cdot -60}{z - t} + a \cdot 120\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+21}:\\ \;\;\;\;\frac{x - y}{\frac{t}{-60}} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{\frac{z}{60}} + a \cdot 120\\ \end{array} \]
Alternative 12
Error24.9
Cost844
\[\begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-44}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-300}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-75}:\\ \;\;\;\;-60 \cdot \frac{x - y}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 13
Error14.9
Cost840
\[\begin{array}{l} \mathbf{if}\;a \leq -9.6 \cdot 10^{-44}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 4 \cdot 10^{-69}:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 14
Error0.2
Cost832
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]
Alternative 15
Error0.4
Cost832
\[\frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120 \]
Alternative 16
Error28.3
Cost716
\[\begin{array}{l} \mathbf{if}\;a \leq -5.1 \cdot 10^{-132}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -1.85 \cdot 10^{-305}:\\ \;\;\;\;-60 \cdot \frac{y}{z}\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-119}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 17
Error24.9
Cost713
\[\begin{array}{l} \mathbf{if}\;a \leq -3.8 \cdot 10^{-44} \lor \neg \left(a \leq 3.8 \cdot 10^{-69}\right):\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \end{array} \]
Alternative 18
Error28.1
Cost584
\[\begin{array}{l} \mathbf{if}\;a \leq -2.7 \cdot 10^{-170}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-119}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 19
Error28.9
Cost192
\[a \cdot 120 \]

Error

Reproduce

herbie shell --seed 2022364 
(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)))