?

Average Error: 0.65% → 0.2%
Time: 23.5s
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

Original0.65%
Target0.23%
Herbie0.2%
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]

Derivation?

  1. Initial program 0.65

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

  2. Simplified0.2

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

    [Start]0.65

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

    +-commutative [=>]0.65

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

    fma-def [=>]0.61

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

    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
Error23.3%
Cost1616
\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -5 \cdot 10^{+43}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq -200000000000:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \mathbf{elif}\;a \cdot 120 \leq -1 \cdot 10^{-5}:\\ \;\;\;\;a \cdot 120 + x \cdot \frac{60}{z}\\ \mathbf{elif}\;a \cdot 120 \leq 5 \cdot 10^{+37}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{-60}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 3
Error44.45%
Cost1505
\[\begin{array}{l} t_1 := 60 \cdot \frac{x}{z - t}\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{+174}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.04 \cdot 10^{+144}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9.6 \cdot 10^{-101}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-189}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{+130} \lor \neg \left(x \leq 5 \cdot 10^{+269}\right) \land x \leq 1.86 \cdot 10^{+298}:\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 4
Error44.43%
Cost1505
\[\begin{array}{l} t_1 := 60 \cdot \frac{x}{z - t}\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{+173}:\\ \;\;\;\;\frac{60}{z - t} \cdot x\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{+144}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;x \leq -9.8 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -6.4 \cdot 10^{-98}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;x \leq -8.2 \cdot 10^{-190}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{elif}\;x \leq 2.4 \cdot 10^{+129} \lor \neg \left(x \leq 6.5 \cdot 10^{+269}\right) \land x \leq 1.95 \cdot 10^{+298}:\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 5
Error44.46%
Cost1505
\[\begin{array}{l} t_1 := 60 \cdot \frac{x}{z - t}\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+173}:\\ \;\;\;\;\frac{60}{z - t} \cdot x\\ \mathbf{elif}\;x \leq -3.9 \cdot 10^{+144}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;x \leq -3.3 \cdot 10^{+107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -9.6 \cdot 10^{-101}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-189}:\\ \;\;\;\;y \cdot \frac{-60}{z - t}\\ \mathbf{elif}\;x \leq 1.45 \cdot 10^{+129} \lor \neg \left(x \leq 6.5 \cdot 10^{+269}\right) \land x \leq 1.86 \cdot 10^{+298}:\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error44.43%
Cost1505
\[\begin{array}{l} t_1 := 60 \cdot \frac{x}{z - t}\\ \mathbf{if}\;x \leq -2 \cdot 10^{+175}:\\ \;\;\;\;\frac{x}{\frac{z - t}{60}}\\ \mathbf{elif}\;x \leq -1.85 \cdot 10^{+144}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{+108}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.8 \cdot 10^{-100}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;x \leq -1.15 \cdot 10^{-189}:\\ \;\;\;\;y \cdot \frac{-60}{z - t}\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{+130} \lor \neg \left(x \leq 6.5 \cdot 10^{+269}\right) \land x \leq 1.86 \cdot 10^{+298}:\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error44.48%
Cost1505
\[\begin{array}{l} \mathbf{if}\;x \leq -8 \cdot 10^{+175}:\\ \;\;\;\;\frac{x}{\frac{z - t}{60}}\\ \mathbf{elif}\;x \leq -7.7 \cdot 10^{+143}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;x \leq -8.5 \cdot 10^{+107}:\\ \;\;\;\;\frac{60 \cdot x}{z - t}\\ \mathbf{elif}\;x \leq -8 \cdot 10^{-101}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;x \leq -6.5 \cdot 10^{-192}:\\ \;\;\;\;y \cdot \frac{-60}{z - t}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{+130} \lor \neg \left(x \leq 6.5 \cdot 10^{+269}\right) \land x \leq 1.86 \cdot 10^{+298}:\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;60 \cdot \frac{x}{z - t}\\ \end{array} \]
Alternative 8
Error45.93%
Cost1376
\[\begin{array}{l} t_1 := -60 \cdot \frac{y}{z}\\ t_2 := 60 \cdot \frac{x}{z}\\ \mathbf{if}\;a \leq -2.7 \cdot 10^{-103}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -8.5 \cdot 10^{-220}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-248}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-184}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 6.3 \cdot 10^{-118}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-101}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{-21}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 9
Error45.89%
Cost1376
\[\begin{array}{l} t_1 := y \cdot \frac{-60}{z}\\ t_2 := 60 \cdot \frac{x}{z}\\ \mathbf{if}\;a \leq -3.2 \cdot 10^{-101}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -2 \cdot 10^{-214}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-249}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7.8 \cdot 10^{-184}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 5.2 \cdot 10^{-118}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 5.6 \cdot 10^{-105}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq 2.35 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{-21}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 10
Error45.97%
Cost1376
\[\begin{array}{l} t_1 := y \cdot \frac{-60}{z}\\ t_2 := 60 \cdot \frac{x}{z}\\ \mathbf{if}\;a \leq -1.1 \cdot 10^{-101}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -8.8 \cdot 10^{-219}:\\ \;\;\;\;x \cdot \frac{60}{z}\\ \mathbf{elif}\;a \leq 3.5 \cdot 10^{-251}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 7.5 \cdot 10^{-184}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 6.3 \cdot 10^{-118}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-103}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-21}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 11
Error45.95%
Cost1376
\[\begin{array}{l} t_1 := y \cdot \frac{-60}{z}\\ t_2 := 60 \cdot \frac{x}{z}\\ \mathbf{if}\;a \leq -1.85 \cdot 10^{-103}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -1.75 \cdot 10^{-215}:\\ \;\;\;\;\frac{60}{\frac{z}{x}}\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{-251}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.5 \cdot 10^{-184}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 6.6 \cdot 10^{-118}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 9.8 \cdot 10^{-102}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq 3.25 \cdot 10^{-46}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-19}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 12
Error45.68%
Cost1376
\[\begin{array}{l} t_1 := y \cdot \frac{-60}{z}\\ t_2 := 60 \cdot \frac{x}{z}\\ \mathbf{if}\;a \leq -3.5 \cdot 10^{-123}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -9 \cdot 10^{-220}:\\ \;\;\;\;\frac{60 \cdot x}{z}\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{-251}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-184}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;a \leq 4.5 \cdot 10^{-118}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 2 \cdot 10^{-102}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{elif}\;a \leq 1.32 \cdot 10^{-45}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 2.6 \cdot 10^{-21}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 13
Error16.35%
Cost1225
\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -4 \cdot 10^{-120} \lor \neg \left(a \cdot 120 \leq 4 \cdot 10^{-29}\right):\\ \;\;\;\;\frac{60}{z - t} \cdot x + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{-60}{z - t}\\ \end{array} \]
Alternative 14
Error39.8%
Cost976
\[\begin{array}{l} t_1 := -60 \cdot \frac{y}{z - t}\\ \mathbf{if}\;a \leq -3.7 \cdot 10^{-99}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 3.1 \cdot 10^{-231}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{-185}:\\ \;\;\;\;60 \cdot \frac{x}{z}\\ \mathbf{elif}\;a \leq 2.7 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 15
Error10.73%
Cost969
\[\begin{array}{l} \mathbf{if}\;x \leq -2.2 \cdot 10^{+107} \lor \neg \left(x \leq 2.8 \cdot 10^{+80}\right):\\ \;\;\;\;\frac{60}{z - t} \cdot x + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot -60}{z - t} + a \cdot 120\\ \end{array} \]
Alternative 16
Error23.28%
Cost841
\[\begin{array}{l} \mathbf{if}\;a \leq -4.2 \cdot 10^{+39} \lor \neg \left(a \leq 1.05 \cdot 10^{+36}\right):\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) \cdot \frac{-60}{z - t}\\ \end{array} \]
Alternative 17
Error23.24%
Cost840
\[\begin{array}{l} \mathbf{if}\;a \leq -9.8 \cdot 10^{+38}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.1 \cdot 10^{+36}:\\ \;\;\;\;60 \cdot \frac{x - y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]
Alternative 18
Error0.23%
Cost832
\[\frac{60}{\frac{z - t}{x - y}} + a \cdot 120 \]
Alternative 19
Error45.1%
Cost585
\[\begin{array}{l} \mathbf{if}\;a \leq -1.26 \cdot 10^{-207} \lor \neg \left(a \leq 3.5 \cdot 10^{-48}\right):\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;-60 \cdot \frac{y}{z}\\ \end{array} \]
Alternative 20
Error45.39%
Cost192
\[a \cdot 120 \]

Error

Reproduce?

herbie shell --seed 2023088 
(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)))