Average Error: 0.5 → 0.1
Time: 18.1s
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

Original0.5
Target0.2
Herbie0.1
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]

Derivation

  1. Initial program 0.5

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

  2. Simplified0.1

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

    [Start]0.5

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

    +-commutative [=>]0.5

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

    fma-def [=>]0.5

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

    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
Error0.1
Cost7104
\[\mathsf{fma}\left(60, \frac{x - y}{z - t}, a \cdot 120\right) \]
Alternative 2
Error10.6
Cost1225
\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -2 \cdot 10^{-28} \lor \neg \left(a \cdot 120 \leq 10^{-91}\right):\\ \;\;\;\;\frac{60}{\frac{z - t}{x}} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;\frac{60}{z - t} \cdot \left(x - y\right)\\ \end{array} \]
Alternative 3
Error25.0
Cost1108
\[\begin{array}{l} t_1 := -60 \cdot \frac{y}{z - t}\\ t_2 := 60 \cdot \frac{x}{z - t}\\ \mathbf{if}\;a \leq -8 \cdot 10^{-35}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-290}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-101}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{-55}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 4
Error25.0
Cost1108
\[\begin{array}{l} t_1 := 60 \cdot \frac{x}{z - t}\\ \mathbf{if}\;a \leq -1.55 \cdot 10^{-34}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 2.45 \cdot 10^{-288}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7.1 \cdot 10^{-101}:\\ \;\;\;\;y \cdot \frac{-60}{z - t}\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{-55}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 5.1 \cdot 10^{-50}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 5
Error25.1
Cost1108
\[\begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{-33}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.75 \cdot 10^{-283}:\\ \;\;\;\;60 \cdot \frac{x}{z - t}\\ \mathbf{elif}\;a \leq 4.8 \cdot 10^{-101}:\\ \;\;\;\;y \cdot \frac{-60}{z - t}\\ \mathbf{elif}\;a \leq 1.16 \cdot 10^{-54}:\\ \;\;\;\;\frac{60}{z - t} \cdot x\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-48}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 6
Error25.1
Cost1108
\[\begin{array}{l} \mathbf{if}\;a \leq -1.05 \cdot 10^{-33}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-283}:\\ \;\;\;\;\frac{x}{\left(z - t\right) \cdot 0.016666666666666666}\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{-102}:\\ \;\;\;\;y \cdot \frac{-60}{z - t}\\ \mathbf{elif}\;a \leq 10^{-55}:\\ \;\;\;\;\frac{60}{z - t} \cdot x\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-50}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 7
Error25.0
Cost1108
\[\begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{-35}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 5.25 \cdot 10^{-281}:\\ \;\;\;\;\frac{x}{\frac{z - t}{60}}\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-101}:\\ \;\;\;\;y \cdot \frac{-60}{z - t}\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-54}:\\ \;\;\;\;\frac{60}{z - t} \cdot x\\ \mathbf{elif}\;a \leq 5.1 \cdot 10^{-50}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 8
Error24.8
Cost1108
\[\begin{array}{l} \mathbf{if}\;a \leq -1.55 \cdot 10^{-34}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -1.46 \cdot 10^{-248}:\\ \;\;\;\;\frac{x - y}{\frac{t}{-60}}\\ \mathbf{elif}\;a \leq 5 \cdot 10^{-283}:\\ \;\;\;\;60 \cdot \frac{x}{z - t}\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{-145}:\\ \;\;\;\;y \cdot \frac{-60}{z - t}\\ \mathbf{elif}\;a \leq 4.7 \cdot 10^{-61}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{60}{z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 9
Error14.7
Cost1097
\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -2 \cdot 10^{-28} \lor \neg \left(a \cdot 120 \leq 5 \cdot 10^{-52}\right):\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;\frac{60}{z - t} \cdot \left(x - y\right)\\ \end{array} \]
Alternative 10
Error6.7
Cost969
\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+61} \lor \neg \left(x \leq 4.5 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{60}{\frac{z - t}{x}} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;\frac{60}{\frac{t - z}{y}} + a \cdot 120\\ \end{array} \]
Alternative 11
Error29.5
Cost848
\[\begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{-35}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-229}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq -1.8 \cdot 10^{-246}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.3 \cdot 10^{-100}:\\ \;\;\;\;-60 \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 12
Error30.2
Cost848
\[\begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{-35}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -3.3 \cdot 10^{-229}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq -2.05 \cdot 10^{-247}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-62}:\\ \;\;\;\;60 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 13
Error29.6
Cost848
\[\begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{-35}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -2.4 \cdot 10^{-229}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq -5.6 \cdot 10^{-235}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.45 \cdot 10^{-121}:\\ \;\;\;\;60 \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 14
Error29.6
Cost848
\[\begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{-35}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -3.05 \cdot 10^{-229}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq -3.2 \cdot 10^{-235}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 2.8 \cdot 10^{-123}:\\ \;\;\;\;y \cdot \frac{60}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 15
Error29.6
Cost848
\[\begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{-35}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -4.7 \cdot 10^{-229}:\\ \;\;\;\;\frac{-60}{\frac{t}{x}}\\ \mathbf{elif}\;a \leq -8 \cdot 10^{-235}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{-123}:\\ \;\;\;\;y \cdot \frac{60}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 16
Error29.6
Cost848
\[\begin{array}{l} \mathbf{if}\;a \leq -8 \cdot 10^{-35}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -3.15 \cdot 10^{-229}:\\ \;\;\;\;\frac{x}{t \cdot -0.016666666666666666}\\ \mathbf{elif}\;a \leq -2.5 \cdot 10^{-236}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.18 \cdot 10^{-122}:\\ \;\;\;\;y \cdot \frac{60}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 17
Error0.2
Cost832
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]
Alternative 18
Error0.5
Cost832
\[\frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120 \]
Alternative 19
Error24.6
Cost712
\[\begin{array}{l} \mathbf{if}\;a \leq -6.5 \cdot 10^{-35}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 2.65 \cdot 10^{-98}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 20
Error29.5
Cost584
\[\begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-35}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.05 \cdot 10^{-162}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 21
Error29.1
Cost192
\[a \cdot 120 \]

Error

Reproduce

herbie shell --seed 2022356 
(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)))