?

Average Error: 0.77% → 0.2%
Time: 22.3s
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

Original0.77%
Target0.24%
Herbie0.2%
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]

Derivation?

  1. Initial program 0.77

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

  2. Simplified0.2

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

    [Start]0.77

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

    +-commutative [=>]0.77

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

    fma-def [=>]0.74

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

    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
Error18.15%
Cost1746
\[\begin{array}{l} t_1 := \frac{60}{z - t}\\ \mathbf{if}\;a \cdot 120 \leq -5 \cdot 10^{-127} \lor \neg \left(a \cdot 120 \leq 10^{-238}\right) \land \left(a \cdot 120 \leq 2 \cdot 10^{-104} \lor \neg \left(a \cdot 120 \leq 2 \cdot 10^{-44}\right)\right):\\ \;\;\;\;a \cdot 120 + t_1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1 \cdot \left(x - y\right)\\ \end{array} \]
Alternative 2
Error18.15%
Cost1745
\[\begin{array}{l} t_1 := \frac{60}{z - t}\\ \mathbf{if}\;a \cdot 120 \leq -5 \cdot 10^{-127}:\\ \;\;\;\;a \cdot 120 + t_1 \cdot x\\ \mathbf{elif}\;a \cdot 120 \leq 10^{-238} \lor \neg \left(a \cdot 120 \leq 2 \cdot 10^{-104}\right) \land a \cdot 120 \leq 2 \cdot 10^{-44}:\\ \;\;\;\;t_1 \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120 + \frac{60}{\frac{z - t}{x}}\\ \end{array} \]
Alternative 3
Error32.47%
Cost1632
\[\begin{array}{l} t_1 := a \cdot 120 + x \cdot \frac{-60}{t}\\ t_2 := a \cdot 120 + \frac{60}{\frac{z}{x}}\\ t_3 := \frac{60}{z - t} \cdot \left(x - y\right)\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+19}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -9.4 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-126}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.35 \cdot 10^{-195}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-276}:\\ \;\;\;\;a \cdot 120 + 60 \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-161}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-53}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 4
Error32.34%
Cost1632
\[\begin{array}{l} t_1 := a \cdot 120 + x \cdot \frac{-60}{t}\\ t_2 := a \cdot 120 + \frac{60}{\frac{z}{x}}\\ t_3 := \frac{60}{z - t} \cdot \left(x - y\right)\\ \mathbf{if}\;z \leq -9 \cdot 10^{+18}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.32 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-126}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-199}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-278}:\\ \;\;\;\;a \cdot 120 + \frac{y}{\frac{t}{60}}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-161}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.6 \cdot 10^{-55}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 5
Error32.56%
Cost1632
\[\begin{array}{l} t_1 := a \cdot 120 + x \cdot \frac{-60}{t}\\ t_2 := a \cdot 120 + \frac{60}{\frac{z}{x}}\\ t_3 := \frac{60}{z - t} \cdot \left(x - y\right)\\ \mathbf{if}\;z \leq -1.02 \cdot 10^{+18}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-75}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.85 \cdot 10^{-126}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq -1.95 \cdot 10^{-196}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-275}:\\ \;\;\;\;a \cdot 120 + \frac{y}{\frac{t}{60}}\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-161}:\\ \;\;\;\;a \cdot 120 + \frac{x \cdot -60}{t}\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-49}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+39}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 6
Error11.47%
Cost1234
\[\begin{array}{l} \mathbf{if}\;y \leq -4.2 \cdot 10^{+207} \lor \neg \left(y \leq -8.5 \cdot 10^{+186} \lor \neg \left(y \leq -2.7 \cdot 10^{+111}\right) \land y \leq 1.6 \cdot 10^{+133}\right):\\ \;\;\;\;a \cdot 120 + \frac{y \cdot -60}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120 + \frac{60}{\frac{z - t}{x}}\\ \end{array} \]
Alternative 7
Error23.23%
Cost1096
\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -2 \cdot 10^{-29}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{60}{z - t} \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 8
Error40.44%
Cost976
\[\begin{array}{l} t_1 := -60 \cdot \frac{y}{z - t}\\ \mathbf{if}\;a \leq -4.6 \cdot 10^{-128}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-222}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-104}:\\ \;\;\;\;\frac{x}{t \cdot -0.016666666666666666}\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 9
Error39.14%
Cost844
\[\begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-128}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.95 \cdot 10^{-290}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-14}:\\ \;\;\;\;60 \cdot \frac{x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 10
Error39.11%
Cost844
\[\begin{array}{l} \mathbf{if}\;a \leq -9.5 \cdot 10^{-128}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.42 \cdot 10^{-291}:\\ \;\;\;\;y \cdot \frac{-60}{z - t}\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-14}:\\ \;\;\;\;60 \cdot \frac{x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 11
Error39.11%
Cost844
\[\begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{-128}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-291}:\\ \;\;\;\;y \cdot \frac{-60}{z - t}\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-14}:\\ \;\;\;\;\frac{60}{z - t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 12
Error39.21%
Cost844
\[\begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{-128}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-289}:\\ \;\;\;\;\frac{-60}{\frac{z - t}{y}}\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{-13}:\\ \;\;\;\;\frac{60}{z - t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 13
Error39.15%
Cost844
\[\begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{-128}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-293}:\\ \;\;\;\;\frac{-60}{\frac{z - t}{y}}\\ \mathbf{elif}\;a \leq 7 \cdot 10^{-13}:\\ \;\;\;\;\frac{x}{\frac{z - t}{60}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 14
Error0.24%
Cost832
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]
Alternative 15
Error43.91%
Cost584
\[\begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-150}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 3.55 \cdot 10^{-127}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 16
Error43.93%
Cost584
\[\begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{-150}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-125}:\\ \;\;\;\;x \cdot \frac{-60}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 17
Error44.18%
Cost584
\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-150}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-110}:\\ \;\;\;\;\frac{x}{t \cdot -0.016666666666666666}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 18
Error45.11%
Cost192
\[a \cdot 120 \]

Error

Reproduce?

herbie shell --seed 2023121 
(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)))