Average Error: 2.0 → 0.3
Time: 23.3s
Precision: binary64
Cost: 7104
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
↓
\[\mathsf{fma}\left(a, \frac{z - y}{\left(1 + t\right) - z}, x\right) \]
(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) (- (+ 1.0 t) 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) / ((1.0 + t) - 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(Float64(1.0 + t) - 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[(N[(1.0 + t), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

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

Error

Target

Original2.0
Target0.3
Herbie0.3
\[x - \frac{y - z}{\left(t - z\right) + 1} \cdot a \]

Derivation

  1. Initial program 2.0

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

  2. Taylor expanded in a around 0 0.3

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

  3. Simplified0.3

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

Alternatives

Alternative 1
Error11.5
Cost1364
\[\begin{array}{l} t_1 := \left(1 + t\right) - z\\ t_2 := a \cdot \frac{z - y}{t_1}\\ \mathbf{if}\;z \leq -8.5 \cdot 10^{+30}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-26}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-7}:\\ \;\;\;\;x - \frac{y \cdot a}{1 + t}\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+46}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{a \cdot z}{t_1}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - y}{\frac{z}{a}}\\ \end{array} \]
Alternative 2
Error11.6
Cost1236
\[\begin{array}{l} t_1 := a \cdot \frac{z - y}{\left(1 + t\right) - z}\\ \mathbf{if}\;z \leq -1.65 \cdot 10^{+30}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-26}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-7}:\\ \;\;\;\;x - \frac{y \cdot a}{1 + t}\\ \mathbf{elif}\;z \leq 2.25 \cdot 10^{+52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+60}:\\ \;\;\;\;x - \left(-\frac{a \cdot y}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - y}{\frac{z}{a}}\\ \end{array} \]
Alternative 3
Error9.6
Cost1100
\[\begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+23}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 6 \cdot 10^{-7}:\\ \;\;\;\;x - \begin{array}{l} \mathbf{if}\;y \ne 0:\\ \;\;\;\;\frac{a}{\frac{1 + t}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot a}{1 + t}\\ \end{array}\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{+45}:\\ \;\;\;\;a \cdot \frac{z - y}{\left(1 + t\right) - z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - y}{\frac{z}{a}}\\ \end{array} \]
Alternative 4
Error11.2
Cost840
\[\begin{array}{l} \mathbf{if}\;z \leq -3.45 \cdot 10^{+22}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+54}:\\ \;\;\;\;x - \frac{y \cdot a}{1 + t}\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]
Alternative 5
Error10.7
Cost840
\[\begin{array}{l} \mathbf{if}\;z \leq -3.2 \cdot 10^{+22}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+54}:\\ \;\;\;\;x - \frac{y \cdot a}{1 + t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z - y}{\frac{z}{a}}\\ \end{array} \]
Alternative 6
Error2.0
Cost832
\[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
Alternative 7
Error17.4
Cost584
\[\begin{array}{l} \mathbf{if}\;z \leq -0.86:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+54}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]
Alternative 8
Error26.7
Cost524
\[\begin{array}{l} \mathbf{if}\;a \leq -3.05 \cdot 10^{+203}:\\ \;\;\;\;-a\\ \mathbf{elif}\;a \leq 2.75 \cdot 10^{+216}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 10^{+296}:\\ \;\;\;\;-a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
Alternative 9
Error19.5
Cost456
\[\begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-32}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+58}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \]
Alternative 10
Error27.6
Cost64
\[x \]

Error

Reproduce

herbie shell --seed 2023010 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :herbie-target
  (- x (* (/ (- y z) (+ (- t z) 1.0)) a))

  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))