\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]

↓

\[\mathsf{fma}\left(x - y, \frac{4}{z}, -2\right) \]

(FPCore (x y z) :precision binary64 (/ (* 4.0 (- (- x y) (* z 0.5))) z))

↓

(FPCore (x y z) :precision binary64 (fma (- x y) (/ 4.0 z) -2.0))

double code(double x, double y, double z) {
	return (4.0 * ((x - y) - (z * 0.5))) / z;
}

↓

double code(double x, double y, double z) {
	return fma((x - y), (4.0 / z), -2.0);
}

function code(x, y, z)
	return Float64(Float64(4.0 * Float64(Float64(x - y) - Float64(z * 0.5))) / z)
end

↓

function code(x, y, z)
	return fma(Float64(x - y), Float64(4.0 / z), -2.0)
end

code[x_, y_, z_] := N[(N[(4.0 * N[(N[(x - y), $MachinePrecision] - N[(z * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(x - y), $MachinePrecision] * N[(4.0 / z), $MachinePrecision] + -2.0), $MachinePrecision]

\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z}

↓

\mathsf{fma}\left(x - y, \frac{4}{z}, -2\right)

Original	0.1
Target	0.0
Herbie	0.2

Derivation

Initial program 0.1
\[\frac{4 \cdot \left(\left(x - y\right) - z \cdot 0.5\right)}{z} \]
Simplified0.2
\[\leadsto \color{blue}{\mathsf{fma}\left(x - y, \frac{4}{z}, -2\right)} \]
Proof
(fma.f64 (-.f64 x y) (/.f64 4 z) -2): 0 points increase in error, 0 points decrease in error
(fma.f64 (-.f64 x y) (/.f64 4 z) (Rewrite<= metadata-eval (*.f64 1 -2))): 0 points increase in error, 0 points decrease in error
(fma.f64 (-.f64 x y) (/.f64 4 z) (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 (neg.f64 z) (neg.f64 z))) -2)): 0 points increase in error, 0 points decrease in error
(fma.f64 (-.f64 x y) (/.f64 4 z) (*.f64 (/.f64 (neg.f64 z) (neg.f64 z)) (Rewrite<= metadata-eval (/.f64 2 -1)))): 0 points increase in error, 0 points decrease in error
(fma.f64 (-.f64 x y) (/.f64 4 z) (*.f64 (/.f64 (neg.f64 z) (neg.f64 z)) (/.f64 (Rewrite<= metadata-eval (*.f64 1/2 4)) -1))): 0 points increase in error, 0 points decrease in error
(fma.f64 (-.f64 x y) (/.f64 4 z) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (neg.f64 z) (*.f64 1/2 4)) (*.f64 (neg.f64 z) -1)))): 0 points increase in error, 0 points decrease in error
(fma.f64 (-.f64 x y) (/.f64 4 z) (/.f64 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (neg.f64 z) 1/2) 4)) (*.f64 (neg.f64 z) -1))): 0 points increase in error, 0 points decrease in error
(fma.f64 (-.f64 x y) (/.f64 4 z) (/.f64 (*.f64 (*.f64 (neg.f64 z) 1/2) 4) (Rewrite=> *-commutative_binary64 (*.f64 -1 (neg.f64 z))))): 0 points increase in error, 0 points decrease in error
(fma.f64 (-.f64 x y) (/.f64 4 z) (/.f64 (*.f64 (*.f64 (neg.f64 z) 1/2) 4) (Rewrite<= neg-mul-1_binary64 (neg.f64 (neg.f64 z))))): 0 points increase in error, 0 points decrease in error
(fma.f64 (-.f64 x y) (/.f64 4 z) (/.f64 (*.f64 (*.f64 (neg.f64 z) 1/2) 4) (Rewrite=> remove-double-neg_binary64 z))): 0 points increase in error, 0 points decrease in error
(fma.f64 (-.f64 x y) (/.f64 4 z) (Rewrite<= associate-*r/_binary64 (*.f64 (*.f64 (neg.f64 z) 1/2) (/.f64 4 z)))): 16 points increase in error, 0 points decrease in error
(Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 x y) (/.f64 4 z)) (*.f64 (*.f64 (neg.f64 z) 1/2) (/.f64 4 z)))): 2 points increase in error, 0 points decrease in error
(Rewrite<= distribute-rgt-in_binary64 (*.f64 (/.f64 4 z) (+.f64 (-.f64 x y) (*.f64 (neg.f64 z) 1/2)))): 10 points increase in error, 2 points decrease in error
(*.f64 (/.f64 4 z) (Rewrite<= cancel-sign-sub-inv_binary64 (-.f64 (-.f64 x y) (*.f64 z 1/2)))): 0 points increase in error, 0 points decrease in error
(Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 4 (-.f64 (-.f64 x y) (*.f64 z 1/2))) z)): 6 points increase in error, 57 points decrease in error
Final simplification0.2
\[\leadsto \mathsf{fma}\left(x - y, \frac{4}{z}, -2\right) \]

Alternatives

Alternative 1
Error	32.7
Cost	1904

\[\begin{array}{l} t_0 := y \cdot \frac{-4}{z}\\ t_1 := 4 \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{+163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{+141}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq -2.0264942626311143 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.210891807363003 \cdot 10^{-185}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq -1.6162590536144845 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.444363808362934 \cdot 10^{-193}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 5.264380306213504 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.4148326891456618 \cdot 10^{-18}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 5.892807328086508 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.480015898889749:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+224}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	32.7
Cost	1904

\[\begin{array}{l} t_0 := \frac{-4}{\frac{z}{y}}\\ t_1 := y \cdot \frac{-4}{z}\\ t_2 := 4 \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{+163}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{+141}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq -2.0264942626311143 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.210891807363003 \cdot 10^{-185}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq -1.6162590536144845 \cdot 10^{-230}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 4.444363808362934 \cdot 10^{-193}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 5.264380306213504 \cdot 10^{-131}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 1.4148326891456618 \cdot 10^{-18}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 5.892807328086508 \cdot 10^{-8}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq 6.480015898889749:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+224}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	32.7
Cost	1904

\[\begin{array}{l} t_0 := -4 \cdot \frac{y}{z}\\ t_1 := 4 \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{+163}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{+141}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq -2.0264942626311143 \cdot 10^{+20}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.210891807363003 \cdot 10^{-185}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq -1.6162590536144845 \cdot 10^{-230}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.444363808362934 \cdot 10^{-193}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 5.264380306213504 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.4148326891456618 \cdot 10^{-18}:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 5.892807328086508 \cdot 10^{-8}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6.480015898889749:\\ \;\;\;\;-2\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{+74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+224}:\\ \;\;\;\;-2\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	9.0
Cost	976

\[\begin{array}{l} t_0 := -2 + -4 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -1 \cdot 10^{+100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -1.487615532185358 \cdot 10^{+58}:\\ \;\;\;\;-4 \cdot \frac{y - x}{z}\\ \mathbf{elif}\;y \leq -8.995287943844655 \cdot 10^{-17}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.892807328086508 \cdot 10^{-8}:\\ \;\;\;\;-2 + 4 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	14.8
Cost	712

\[\begin{array}{l} t_0 := -4 \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+200}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+224}:\\ \;\;\;\;-2 + 4 \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	11.4
Cost	712

\[\begin{array}{l} t_0 := -2 + 4 \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1.2327914021946877 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.5060208051603787 \cdot 10^{+68}:\\ \;\;\;\;-4 \cdot \frac{y - x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	29.6
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -3.4811650262834927 \cdot 10^{+51}:\\ \;\;\;\;-2\\ \mathbf{elif}\;z \leq 6.510188501213909 \cdot 10^{+70}:\\ \;\;\;\;y \cdot \frac{-4}{z}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

Alternative 8
Error	0.2
Cost	576

\[-2 + \left(x - y\right) \cdot \frac{4}{z} \]

Alternative 9
Error	37.1
Cost	64

\[-2 \]

Error

Target

Derivation

Alternatives

Error

Reproduce