\[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]

↓

\[x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)} \]

(FPCore (x y z)
 :precision binary64
 (+ x (/ y (- (* 1.1283791670955126 (exp z)) (* x y)))))

↓

(FPCore (x y z)
 :precision binary64
 (+ x (/ -1.0 (fma (exp z) (/ -1.1283791670955126 y) x))))

double code(double x, double y, double z) {
	return x + (y / ((1.1283791670955126 * exp(z)) - (x * y)));
}

↓

double code(double x, double y, double z) {
	return x + (-1.0 / fma(exp(z), (-1.1283791670955126 / y), x));
}

function code(x, y, z)
	return Float64(x + Float64(y / Float64(Float64(1.1283791670955126 * exp(z)) - Float64(x * y))))
end

↓

function code(x, y, z)
	return Float64(x + Float64(-1.0 / fma(exp(z), Float64(-1.1283791670955126 / y), x)))
end

code[x_, y_, z_] := N[(x + N[(y / N[(N[(1.1283791670955126 * N[Exp[z], $MachinePrecision]), $MachinePrecision] - N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_] := N[(x + N[(-1.0 / N[(N[Exp[z], $MachinePrecision] * N[(-1.1283791670955126 / y), $MachinePrecision] + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y}

↓

x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)}

Original	2.9
Target	0.0
Herbie	0.0

Derivation

Initial program 2.9
\[x + \frac{y}{1.1283791670955126 \cdot e^{z} - x \cdot y} \]
Simplified0.0
\[\leadsto \color{blue}{x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)}} \]
Proof
(+.f64 x (/.f64 -1 (fma.f64 (exp.f64 z) (/.f64 -5641895835477563/5000000000000000 y) x))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (fma.f64 (exp.f64 z) (/.f64 (Rewrite<= metadata-eval (neg.f64 5641895835477563/5000000000000000)) y) x))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (fma.f64 (exp.f64 z) (Rewrite<= distribute-neg-frac_binary64 (neg.f64 (/.f64 5641895835477563/5000000000000000 y))) x))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (exp.f64 z) (neg.f64 (/.f64 5641895835477563/5000000000000000 y))) x)))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (+.f64 (*.f64 (exp.f64 z) (Rewrite=> distribute-neg-frac_binary64 (/.f64 (neg.f64 5641895835477563/5000000000000000) y))) x))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (+.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (exp.f64 z) (neg.f64 5641895835477563/5000000000000000)) y)) x))): 1 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (+.f64 (/.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (exp.f64 z) 5641895835477563/5000000000000000))) y) x))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (+.f64 (/.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)))) y) x))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (Rewrite<= +-commutative_binary64 (+.f64 x (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (+.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 x 1)) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (+.f64 (*.f64 x (Rewrite<= metadata-eval (neg.f64 -1))) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (Rewrite<= fma-udef_binary64 (fma.f64 x (neg.f64 -1) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (fma.f64 (Rewrite<= /-rgt-identity_binary64 (/.f64 x 1)) (neg.f64 -1) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (Rewrite<= metadata-eval (neg.f64 -1))) (neg.f64 -1) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (Rewrite=> metadata-eval 1) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (Rewrite<= *-inverses_binary64 (/.f64 y y)) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (/.f64 y y) (Rewrite<= *-lft-identity_binary64 (*.f64 1 (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (/.f64 y y) (*.f64 (Rewrite<= metadata-eval (/.f64 -1 -1)) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (/.f64 y y) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)))) (*.f64 -1 y)))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (/.f64 y y) (/.f64 (*.f64 -1 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)))) (Rewrite<= neg-mul-1_binary64 (neg.f64 y)))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (/.f64 y y) (Rewrite<= associate-*r/_binary64 (*.f64 -1 (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) (neg.f64 y))))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (fma.f64 (/.f64 x (neg.f64 -1)) (/.f64 y y) (Rewrite<= neg-mul-1_binary64 (neg.f64 (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) (neg.f64 y))))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (/.f64 x (neg.f64 -1)) (/.f64 y y)) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) (neg.f64 y)))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (-.f64 (*.f64 (/.f64 x (Rewrite=> metadata-eval 1)) (/.f64 y y)) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) (neg.f64 y))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (-.f64 (*.f64 (Rewrite=> /-rgt-identity_binary64 x) (/.f64 y y)) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) (neg.f64 y))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (-.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x y) y)) (/.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) (neg.f64 y))))): 20 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 x y) y) (/.f64 (Rewrite=> distribute-lft-neg-in_binary64 (*.f64 (neg.f64 5641895835477563/5000000000000000) (exp.f64 z))) (neg.f64 y))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 x y) y) (/.f64 (*.f64 (neg.f64 5641895835477563/5000000000000000) (exp.f64 z)) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 y)))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 x y) y) (Rewrite=> times-frac_binary64 (*.f64 (/.f64 (neg.f64 5641895835477563/5000000000000000) -1) (/.f64 (exp.f64 z) y)))))): 4 points increase in error, 2 points decrease in error
(+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 x y) y) (*.f64 (/.f64 (Rewrite=> metadata-eval -5641895835477563/5000000000000000) -1) (/.f64 (exp.f64 z) y))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 x y) y) (*.f64 (Rewrite=> metadata-eval 5641895835477563/5000000000000000) (/.f64 (exp.f64 z) y))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 -1 (-.f64 (/.f64 (*.f64 x y) y) (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) y))))): 2 points increase in error, 4 points decrease in error
(+.f64 x (/.f64 -1 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (*.f64 x y) (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) y)))): 0 points increase in error, 1 points decrease in error
(+.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 -1 y) (-.f64 (*.f64 x y) (*.f64 5641895835477563/5000000000000000 (exp.f64 z)))))): 8 points increase in error, 7 points decrease in error
(+.f64 x (/.f64 (*.f64 -1 y) (Rewrite=> sub-neg_binary64 (+.f64 (*.f64 x y) (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 (*.f64 -1 y) (Rewrite=> +-commutative_binary64 (+.f64 (neg.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z))) (*.f64 x y))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 (*.f64 -1 y) (+.f64 (Rewrite=> neg-sub0_binary64 (-.f64 0 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)))) (*.f64 x y)))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 (*.f64 -1 y) (Rewrite=> associate-+l-_binary64 (-.f64 0 (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 (*.f64 -1 y) (Rewrite=> sub0-neg_binary64 (neg.f64 (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (/.f64 (*.f64 -1 y) (Rewrite=> neg-mul-1_binary64 (*.f64 -1 (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (Rewrite=> times-frac_binary64 (*.f64 (/.f64 -1 -1) (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y)))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (*.f64 (Rewrite=> metadata-eval 1) (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y))))): 0 points increase in error, 0 points decrease in error
(+.f64 x (Rewrite=> *-lft-identity_binary64 (/.f64 y (-.f64 (*.f64 5641895835477563/5000000000000000 (exp.f64 z)) (*.f64 x y))))): 0 points increase in error, 0 points decrease in error
Final simplification0.0
\[\leadsto x + \frac{-1}{\mathsf{fma}\left(e^{z}, \frac{-1.1283791670955126}{y}, x\right)} \]

Alternatives

Alternative 1
Error	0.1
Cost	20168

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{e^{z} \cdot 1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	0.2
Cost	13896

\[\begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;e^{z} \leq 2:\\ \;\;\;\;x + \frac{y}{\left(1.1283791670955126 + z \cdot 1.1283791670955126\right) - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	19.0
Cost	1248

\[\begin{array}{l} \mathbf{if}\;x \leq -101.39954436873411:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.2317180795851559 \cdot 10^{-82}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq -4.612393584306038 \cdot 10^{-114}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.968297852353123 \cdot 10^{-166}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq -6.100750298429679 \cdot 10^{-245}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.0116244545429168 \cdot 10^{-242}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{elif}\;x \leq 1.7201879346104073 \cdot 10^{-176}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.0143727352669685 \cdot 10^{-144}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	18.2
Cost	1116

\[\begin{array}{l} t_0 := x + \frac{y}{1.1283791670955126}\\ \mathbf{if}\;x \leq -101.39954436873411:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.5825671657737962 \cdot 10^{-75}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq -4.612393584306038 \cdot 10^{-114}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq -2.968297852353123 \cdot 10^{-166}:\\ \;\;\;\;\frac{-1}{x}\\ \mathbf{elif}\;x \leq 1.0116244545429168 \cdot 10^{-242}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 1.7201879346104073 \cdot 10^{-176}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.0143727352669685 \cdot 10^{-144}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	0.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -2219852834.788476:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 1.1647600374701823 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{-1}{x + \frac{-1.1283791670955126}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	0.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -2219852834.788476:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 1.1647600374701823 \cdot 10^{-5}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126 - x \cdot y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	19.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -6.100750298429679 \cdot 10^{-245}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.0116244545429168 \cdot 10^{-242}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{elif}\;x \leq 1.7201879346104073 \cdot 10^{-176}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.0143727352669685 \cdot 10^{-144}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	19.2
Cost	720

\[\begin{array}{l} \mathbf{if}\;x \leq -6.100750298429679 \cdot 10^{-245}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.0116244545429168 \cdot 10^{-242}:\\ \;\;\;\;\frac{y}{1.1283791670955126}\\ \mathbf{elif}\;x \leq 1.7201879346104073 \cdot 10^{-176}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.0143727352669685 \cdot 10^{-144}:\\ \;\;\;\;y \cdot 0.8862269254527579\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	8.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -1.3418398958518488 \cdot 10^{-21}:\\ \;\;\;\;x - \frac{1}{x}\\ \mathbf{elif}\;z \leq 5.662977502530943 \cdot 10^{-42}:\\ \;\;\;\;x + \frac{y}{1.1283791670955126}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Error	20.1
Cost	64

\[x \]

Error

Target

Derivation

Alternatives

Error

Reproduce