?

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

↓

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

(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)

Original	0.4
Target	0.2
Herbie	0.1

Derivation?

Initial program 0.4
\[\frac{60 \cdot \left(x - y\right)}{z - t} + a \cdot 120 \]

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

Final simplification0.1
\[\leadsto \mathsf{fma}\left(a, 120, \frac{60}{z - t} \cdot \left(x - y\right)\right) \]

Alternatives

Alternative 1
Error	0.1
Cost	7104

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

Alternative 2
Error	29.5
Cost	1504

\[\begin{array}{l} t_1 := 60 \cdot \frac{x}{z - t}\\ t_2 := -60 \cdot \frac{y}{z - t}\\ \mathbf{if}\;y \leq -1.7 \cdot 10^{+231}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{+81}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;y \leq -64000000000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{-278}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{-271}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.46 \cdot 10^{-192}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 10^{+116}:\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 3
Error	29.6
Cost	1504

\[\begin{array}{l} t_1 := 60 \cdot \frac{x}{z - t}\\ t_2 := -60 \cdot \frac{y}{z - t}\\ \mathbf{if}\;y \leq -5.4 \cdot 10^{+231}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -1.46 \cdot 10^{+82}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;y \leq -1600000000000:\\ \;\;\;\;\left(x - y\right) \cdot \frac{-60}{t}\\ \mathbf{elif}\;y \leq -1.22 \cdot 10^{-278}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-272}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.46 \cdot 10^{-192}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;y \leq 1.72 \cdot 10^{-180}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 8 \cdot 10^{+116}:\\ \;\;\;\;a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 4
Error	25.9
Cost	1372

\[\begin{array}{l} t_1 := \left(x - y\right) \cdot \frac{-60}{t}\\ \mathbf{if}\;a \leq -5 \cdot 10^{-42}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -1.22 \cdot 10^{-130}:\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{elif}\;a \leq -7.7 \cdot 10^{-177}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq -5.8 \cdot 10^{-308}:\\ \;\;\;\;60 \cdot \frac{x}{z - t}\\ \mathbf{elif}\;a \leq 8.2 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;a \leq 1.02 \cdot 10^{-61}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{60}{z}\\ \mathbf{elif}\;a \leq 8.8 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 5
Error	10.9
Cost	1225

\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -1 \cdot 10^{-174} \lor \neg \left(a \cdot 120 \leq 2 \cdot 10^{-19}\right):\\ \;\;\;\;\frac{y}{\frac{z - t}{-60}} + a \cdot 120\\ \mathbf{else}:\\ \;\;\;\;\frac{60}{z - t} \cdot \left(x - y\right)\\ \end{array} \]

Alternative 6
Error	15.0
Cost	1096

\[\begin{array}{l} \mathbf{if}\;a \cdot 120 \leq -1000000000000:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \cdot 120 \leq 0.001:\\ \;\;\;\;\frac{60}{z - t} \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 7
Error	28.0
Cost	978

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \cdot 10^{+229} \lor \neg \left(y \leq -1.46 \cdot 10^{+82} \lor \neg \left(y \leq -66000000000000\right) \land y \leq 2.6 \cdot 10^{+116}\right):\\ \;\;\;\;-60 \cdot \frac{y}{z - t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 8
Error	0.2
Cost	832

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

Alternative 9
Error	28.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;a \leq -6.8 \cdot 10^{-178}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 6.5 \cdot 10^{-97}:\\ \;\;\;\;-60 \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 10
Error	28.5
Cost	584

\[\begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-177}:\\ \;\;\;\;a \cdot 120\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{-93}:\\ \;\;\;\;x \cdot \frac{-60}{t}\\ \mathbf{else}:\\ \;\;\;\;a \cdot 120\\ \end{array} \]

Alternative 11
Error	28.8
Cost	192

\[a \cdot 120 \]

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Error?

Target

Derivation?

Alternatives

Error

Reproduce?