?

Average Error: 10.5 → 1.3
Time: 11.9s
Precision: binary64
Cost: 6976

?

\[x + \frac{y \cdot \left(z - t\right)}{z - a} \]
↓
\[\mathsf{fma}\left(y, \frac{z - t}{z - a}, x\right) \]
(FPCore (x y z t a) :precision binary64 (+ x (/ (* y (- z t)) (- z a))))
↓
(FPCore (x y z t a) :precision binary64 (fma y (/ (- z t) (- z a)) x))
double code(double x, double y, double z, double t, double a) {
	return x + ((y * (z - t)) / (z - a));
}
↓
double code(double x, double y, double z, double t, double a) {
	return fma(y, ((z - t) / (z - a)), x);
}

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

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

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

Error?

Target

Original10.5
Target1.2
Herbie1.3
\[x + \frac{y}{\frac{z - a}{z - t}} \]

Derivation?

  1. Initial program 10.5

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

  2. Simplified1.3

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

    [Start]10.5

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

    +-commutative [=>]10.5

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

    associate-*r/ [<=]1.3

    \[ \color{blue}{y \cdot \frac{z - t}{z - a}} + x \]

    fma-def [=>]1.3

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

  3. Final simplification1.3

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

Alternatives

Alternative 1
Error14.9
Cost1109
\[\begin{array}{l} t_1 := x - t \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -5.8 \cdot 10^{+103}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{-37}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-22}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+89} \lor \neg \left(z \leq 3.8 \cdot 10^{+168}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 2
Error14.5
Cost1109
\[\begin{array}{l} \mathbf{if}\;z \leq -1.8 \cdot 10^{+103}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-36}:\\ \;\;\;\;x - t \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-22}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+88} \lor \neg \left(z \leq 1.55 \cdot 10^{+123}\right):\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{t}{\frac{z}{y}}\\ \end{array} \]
Alternative 3
Error15.5
Cost1108
\[\begin{array}{l} t_1 := x + \frac{t}{\frac{a}{y}}\\ \mathbf{if}\;z \leq -2.25 \cdot 10^{-66}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 3.2 \cdot 10^{-24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+87}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 7.4 \cdot 10^{+111}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+168}:\\ \;\;\;\;y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 4
Error15.4
Cost1108
\[\begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-38}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{-25}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{+87}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+114}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+168}:\\ \;\;\;\;y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 5
Error15.8
Cost1108
\[\begin{array}{l} \mathbf{if}\;z \leq -3 \cdot 10^{-36}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 4.4 \cdot 10^{-25}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{+87}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+102}:\\ \;\;\;\;\frac{-t}{\frac{z - a}{y}}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+168}:\\ \;\;\;\;y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 6
Error15.8
Cost1108
\[\begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-42}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-25}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{+87}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{+102}:\\ \;\;\;\;\frac{-y}{\frac{z - a}{t}}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+168}:\\ \;\;\;\;y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + x\\ \end{array} \]
Alternative 7
Error23.4
Cost912
\[\begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{+235}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq -3.7 \cdot 10^{-120}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq -9.8 \cdot 10^{-182}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+237}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{-t}{z}\\ \end{array} \]
Alternative 8
Error23.5
Cost912
\[\begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{+232}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;t \leq -1.46 \cdot 10^{-118}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+237}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z}{t}}\\ \end{array} \]
Alternative 9
Error24.8
Cost912
\[\begin{array}{l} \mathbf{if}\;t \leq -3.4 \cdot 10^{+136}:\\ \;\;\;\;y \cdot \left(1 - \frac{t}{z}\right)\\ \mathbf{elif}\;t \leq -3.7 \cdot 10^{-120}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;t \leq -1.5 \cdot 10^{-181}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+237}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-y}{\frac{z}{t}}\\ \end{array} \]
Alternative 10
Error12.2
Cost841
\[\begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-49} \lor \neg \left(z \leq 8.8 \cdot 10^{-23}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \end{array} \]
Alternative 11
Error14.6
Cost840
\[\begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{+136}:\\ \;\;\;\;y \cdot \frac{-t}{z - a}\\ \mathbf{elif}\;t \leq 1.4 \cdot 10^{+160}:\\ \;\;\;\;x + y \cdot \frac{z}{z - a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{z}\\ \end{array} \]
Alternative 12
Error3.0
Cost704
\[x + \left(z - t\right) \cdot \frac{y}{z - a} \]
Alternative 13
Error1.2
Cost704
\[x + \frac{y}{\frac{z - a}{z - t}} \]
Alternative 14
Error19.9
Cost456
\[\begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+189}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{+161}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 15
Error28.6
Cost64
\[x \]

Error

Reproduce?

herbie shell --seed 2023018 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTicks from plot-0.2.3.4, A"
  :precision binary64

  :herbie-target
  (+ x (/ y (/ (- z a) (- z t))))

  (+ x (/ (* y (- z t)) (- z a))))