Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Average Error: 0.0 → 0.0

Time: 8.8s

Precision: binary64

Cost: 6848

Math

\[x + \left(y - z\right) \cdot \left(t - x\right) \]

↓

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

FPCore

(FPCore (x y z t) :precision binary64 (+ x (* (- y z) (- t x))))

↓

(FPCore (x y z t) :precision binary64 (fma (- y z) (- t x) x))

C

double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - x));
}

↓

double code(double x, double y, double z, double t) {
	return fma((y - z), (t - x), x);
}

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y - z) * Float64(t - x)))
end

↓

function code(x, y, z, t)
	return fma(Float64(y - z), Float64(t - x), x)
end

Wolfram

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

↓

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

Target

Original	0.0
Target	0.0
Herbie	0.0

\[x + \left(t \cdot \left(y - z\right) + \left(-x\right) \cdot \left(y - z\right)\right) \]

Derivation

1. Initial program 0.0

   \[x + \left(y - z\right) \cdot \left(t - x\right) \]

2. Simplified 0.0

   \[\leadsto \color{blue}{\mathsf{fma}\left(y - z, t - x, x\right)} \]
   Proof
   (fma.f64 (-.f64 y z) (-.f64 t x) x): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (-.f64 y z) (-.f64 t x)) x)): 2 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 x (*.f64 (-.f64 y z) (-.f64 t x)))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.0

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

Alternatives

Alternative 1
Error	28.6
Cost	2400

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot t\\ t_2 := z \cdot \left(x - t\right)\\ t_3 := x \cdot \left(z + 1\right)\\ t_4 := y \cdot \left(t - x\right)\\ \mathbf{if}\;y - z \leq -4 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y - z \leq 5 \cdot 10^{-154}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y - z \leq 5 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y - z \leq 2 \cdot 10^{-11}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y - z \leq 4 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y - z \leq 5 \cdot 10^{+137}:\\ \;\;\;\;t_4\\ \mathbf{elif}\;y - z \leq 2 \cdot 10^{+155}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y - z \leq 5 \cdot 10^{+157}:\\ \;\;\;\;t_4\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	30.0
Cost	1880

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot t\\ \mathbf{if}\;y - z \leq -5 \cdot 10^{+229}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y - z \leq -5 \cdot 10^{+159}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;y - z \leq -5 \cdot 10^{-50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y - z \leq 5 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;y - z \leq 5 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y - z \leq 2 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Error	28.9
Cost	1880

\[\begin{array}{l} t_1 := \left(y - z\right) \cdot t\\ t_2 := z \cdot \left(x - t\right)\\ \mathbf{if}\;y - z \leq -4 \cdot 10^{-6}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y - z \leq 5 \cdot 10^{-154}:\\ \;\;\;\;x\\ \mathbf{elif}\;y - z \leq 5 \cdot 10^{-116}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y - z \leq 2 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{elif}\;y - z \leq 5 \cdot 10^{+137}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y - z \leq 2 \cdot 10^{+155}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	28.8
Cost	1880

\[\begin{array}{l} t_1 := x \cdot \left(z + 1\right)\\ t_2 := \left(y - z\right) \cdot t\\ t_3 := z \cdot \left(x - t\right)\\ \mathbf{if}\;y - z \leq -4 \cdot 10^{-6}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;y - z \leq 5 \cdot 10^{-154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y - z \leq 5 \cdot 10^{-116}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y - z \leq 2 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y - z \leq 5 \cdot 10^{+137}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y - z \leq 2 \cdot 10^{+155}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	38.4
Cost	1048

\[\begin{array}{l} t_1 := z \cdot \left(-t\right)\\ \mathbf{if}\;z \leq -2.9 \cdot 10^{+162}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.25 \cdot 10^{+134}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{-13}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{-39}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+222}:\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	11.4
Cost	976

\[\begin{array}{l} t_1 := x \cdot \left(\left(z + 1\right) - y\right)\\ t_2 := x + \left(y - z\right) \cdot t\\ \mathbf{if}\;t \leq -1.62 \cdot 10^{-69}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.62 \cdot 10^{-149}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -5.5 \cdot 10^{-190}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.55 \cdot 10^{-113}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 7
Error	17.8
Cost	848

\[\begin{array}{l} t_1 := y \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -1.8 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-26}:\\ \;\;\;\;x \cdot \left(\left(z + 1\right) - y\right)\\ \mathbf{elif}\;y \leq -4.4 \cdot 10^{-60}:\\ \;\;\;\;\left(y - z\right) \cdot t\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-9}:\\ \;\;\;\;x - z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 8
Error	10.6
Cost	844

\[\begin{array}{l} t_1 := z \cdot \left(x - t\right)\\ \mathbf{if}\;z \leq -3.7 \cdot 10^{-10}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.3 \cdot 10^{-119}:\\ \;\;\;\;x + y \cdot \left(t - x\right)\\ \mathbf{elif}\;z \leq 175:\\ \;\;\;\;x + \left(y - z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	17.4
Cost	584

\[\begin{array}{l} t_1 := y \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -0.02:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-10}:\\ \;\;\;\;x - z \cdot t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	0.0
Cost	576

\[x + \left(y - z\right) \cdot \left(t - x\right) \]

Alternative 11
Error	38.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{-58}:\\ \;\;\;\;y \cdot t\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-9}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\\ \end{array} \]

Alternative 12
Error	37.8
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-10}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\\ \end{array} \]

Alternative 13
Error	47.4
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022325 
(FPCore (x y z t)
  :name "Data.Metrics.Snapshot:quantile from metrics-0.3.0.2"
  :precision binary64

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

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