\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]

↓

\[\mathsf{fma}\left(a, \frac{z - y}{t + \left(1 - z\right)}, x\right) \]

(FPCore (x y z t a)
 :precision binary64
 (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))

↓

(FPCore (x y z t a) :precision binary64 (fma a (/ (- z y) (+ t (- 1.0 z))) x))

double code(double x, double y, double z, double t, double a) {
	return x - ((y - z) / (((t - z) + 1.0) / a));
}

↓

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

function code(x, y, z, t, a)
	return Float64(x - Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a)))
end

↓

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

code[x_, y_, z_, t_, a_] := N[(x - N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_, a_] := N[(a * N[(N[(z - y), $MachinePrecision] / N[(t + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}

↓

\mathsf{fma}\left(a, \frac{z - y}{t + \left(1 - z\right)}, x\right)

Original	1.9
Target	0.2
Herbie	0.2

Derivation

Initial program 1.9
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
Simplified0.2
\[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{z - y}{t - \left(z + -1\right)}, x\right)} \]
Proof
(fma.f64 a (/.f64 (-.f64 z y) (-.f64 t (+.f64 z -1))) x): 0 points increase in error, 0 points decrease in error
(fma.f64 a (/.f64 (-.f64 z y) (-.f64 t (+.f64 z (Rewrite<= metadata-eval (neg.f64 1))))) x): 0 points increase in error, 0 points decrease in error
(fma.f64 a (/.f64 (-.f64 z y) (-.f64 t (Rewrite<= sub-neg_binary64 (-.f64 z 1)))) x): 0 points increase in error, 0 points decrease in error
(fma.f64 a (/.f64 (-.f64 z y) (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 t z) 1))) x): 0 points increase in error, 0 points decrease in error
(fma.f64 a (Rewrite=> div-sub_binary64 (-.f64 (/.f64 z (+.f64 (-.f64 t z) 1)) (/.f64 y (+.f64 (-.f64 t z) 1)))) x): 2 points increase in error, 0 points decrease in error
(fma.f64 a (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 z (+.f64 (-.f64 t z) 1)) (neg.f64 (/.f64 y (+.f64 (-.f64 t z) 1))))) x): 0 points increase in error, 0 points decrease in error
(fma.f64 a (+.f64 (Rewrite<= remove-double-neg_binary64 (neg.f64 (neg.f64 (/.f64 z (+.f64 (-.f64 t z) 1))))) (neg.f64 (/.f64 y (+.f64 (-.f64 t z) 1)))) x): 0 points increase in error, 0 points decrease in error
(fma.f64 a (Rewrite<= distribute-neg-in_binary64 (neg.f64 (+.f64 (neg.f64 (/.f64 z (+.f64 (-.f64 t z) 1))) (/.f64 y (+.f64 (-.f64 t z) 1))))) x): 0 points increase in error, 0 points decrease in error
(fma.f64 a (neg.f64 (Rewrite<= +-commutative_binary64 (+.f64 (/.f64 y (+.f64 (-.f64 t z) 1)) (neg.f64 (/.f64 z (+.f64 (-.f64 t z) 1)))))) x): 0 points increase in error, 0 points decrease in error
(fma.f64 a (neg.f64 (Rewrite<= sub-neg_binary64 (-.f64 (/.f64 y (+.f64 (-.f64 t z) 1)) (/.f64 z (+.f64 (-.f64 t z) 1))))) x): 0 points increase in error, 0 points decrease in error
(fma.f64 a (neg.f64 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 y z) (+.f64 (-.f64 t z) 1)))) x): 0 points increase in error, 2 points decrease in error
(Rewrite<= fma-def_binary64 (+.f64 (*.f64 a (neg.f64 (/.f64 (-.f64 y z) (+.f64 (-.f64 t z) 1)))) x)): 1 points increase in error, 0 points decrease in error
(+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 a (/.f64 (-.f64 y z) (+.f64 (-.f64 t z) 1))))) x): 0 points increase in error, 0 points decrease in error
(+.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 (/.f64 (-.f64 y z) (+.f64 (-.f64 t z) 1)) a))) x): 0 points increase in error, 0 points decrease in error
(+.f64 (neg.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) 1) a)))) x): 28 points increase in error, 10 points decrease in error
(Rewrite<= +-commutative_binary64 (+.f64 x (neg.f64 (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) 1) a))))): 0 points increase in error, 0 points decrease in error
(Rewrite<= sub-neg_binary64 (-.f64 x (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) 1) a)))): 0 points increase in error, 0 points decrease in error
Final simplification0.2
\[\leadsto \mathsf{fma}\left(a, \frac{z - y}{t + \left(1 - z\right)}, x\right) \]

Alternatives

Alternative 1
Error	15.5
Cost	1104

\[\begin{array}{l} \mathbf{if}\;t \leq -9.11545711165471 \cdot 10^{+138}:\\ \;\;\;\;x - \frac{a}{\frac{t}{y}}\\ \mathbf{elif}\;t \leq 3.2994064454616784 \cdot 10^{-138}:\\ \;\;\;\;x - \frac{a}{1 - \frac{1}{z}}\\ \mathbf{elif}\;t \leq 1.892173196674624 \cdot 10^{-53}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{elif}\;t \leq 8.254347326089875 \cdot 10^{+107}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - y}{\frac{t}{a}}\\ \end{array} \]

Alternative 2
Error	16.3
Cost	976

\[\begin{array}{l} t_1 := x - \frac{a}{\frac{t}{y}}\\ \mathbf{if}\;t \leq -9.11545711165471 \cdot 10^{+138}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.2994064454616784 \cdot 10^{-138}:\\ \;\;\;\;x - \frac{a}{1 - \frac{1}{z}}\\ \mathbf{elif}\;t \leq 1.892173196674624 \cdot 10^{-53}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{elif}\;t \leq 4.884916956036249 \cdot 10^{+104}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	6.7
Cost	972

\[\begin{array}{l} \mathbf{if}\;t \leq -9.11545711165471 \cdot 10^{+138}:\\ \;\;\;\;x - \frac{a}{\frac{t}{y}}\\ \mathbf{elif}\;t \leq 122.57594854983186:\\ \;\;\;\;x + a \cdot \frac{z - y}{1 - z}\\ \mathbf{elif}\;t \leq 8.689479256336787 \cdot 10^{+196}:\\ \;\;\;\;x - \frac{a}{1 - \frac{t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - y}{\frac{t}{a}}\\ \end{array} \]

Alternative 4
Error	7.5
Cost	904

\[\begin{array}{l} t_1 := x - \frac{a}{\frac{-z}{y - z}}\\ \mathbf{if}\;z \leq -0.006195692644150916:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 141236470609.5425:\\ \;\;\;\;x - \frac{a}{\frac{t + 1}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	7.5
Cost	904

\[\begin{array}{l} t_1 := x + a \cdot \frac{z - y}{-z}\\ \mathbf{if}\;z \leq -0.006195692644150916:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 141236470609.5425:\\ \;\;\;\;x - \frac{a}{\frac{t + 1}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	7.1
Cost	904

\[\begin{array}{l} \mathbf{if}\;z \leq -0.006195692644150916:\\ \;\;\;\;x - \frac{a}{1 + \frac{-1 - t}{z}}\\ \mathbf{elif}\;z \leq 141236470609.5425:\\ \;\;\;\;x - \frac{a}{\frac{t + 1}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + a \cdot \frac{z - y}{-z}\\ \end{array} \]

Alternative 7
Error	9.8
Cost	840

\[\begin{array}{l} t_1 := x - \frac{a}{1 - \frac{1}{z}}\\ \mathbf{if}\;z \leq -0.006195692644150916:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.599496647557363 \cdot 10^{-7}:\\ \;\;\;\;x - \frac{a}{\frac{t + 1}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	7.8
Cost	840

\[\begin{array}{l} t_1 := x - \frac{a}{1 - \frac{t}{z}}\\ \mathbf{if}\;z \leq -3.0475727880824724 \cdot 10^{+65}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.5793985222201038 \cdot 10^{-38}:\\ \;\;\;\;x - \frac{a}{\frac{t + 1}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	0.3
Cost	832

\[x - \frac{a}{\frac{\left(t - z\right) + 1}{y - z}} \]

Alternative 10
Error	0.2
Cost	832

\[x + a \cdot \frac{z - y}{\left(t - z\right) + 1} \]

Alternative 11
Error	16.9
Cost	584

\[\begin{array}{l} \mathbf{if}\;z \leq -0.006195692644150916:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4.599496647557363 \cdot 10^{-7}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 12
Error	19.9
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -3.0475727880824724 \cdot 10^{+65}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 4.599496647557363 \cdot 10^{-7}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]

Alternative 13
Error	27.7
Cost	260

\[\begin{array}{l} \mathbf{if}\;a \leq 1.02 \cdot 10^{+153}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-a\\ \end{array} \]

Alternative 14
Error	28.0
Cost	64

\[x \]

Error

Target

Derivation

Alternatives

Error

Reproduce