Rosa's Benchmark

Average Error: 0.2 → 0.2

Time: 6.4s

Precision: binary64

Cost: 13184

Math

\[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

↓

\[\mathsf{fma}\left({x}^{3}, -0.12900613773279798, x \cdot 0.954929658551372\right) \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))

↓

(FPCore (x)
 :precision binary64
 (fma (pow x 3.0) -0.12900613773279798 (* x 0.954929658551372)))

C

double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

↓

double code(double x) {
	return fma(pow(x, 3.0), -0.12900613773279798, (x * 0.954929658551372));
}

Julia

function code(x)
	return Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(Float64(x * x) * x)))
end

↓

function code(x)
	return fma((x ^ 3.0), -0.12900613773279798, Float64(x * 0.954929658551372))
end

Wolfram

code[x_] := N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[Power[x, 3.0], $MachinePrecision] * -0.12900613773279798 + N[(x * 0.954929658551372), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 0.2

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

2. Simplified 0.2

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.12900613773279798, 0.954929658551372 \cdot x\right)} \]
   Proof
   (fma.f64 (pow.f64 x 3) -6450306886639899/50000000000000000 (*.f64 238732414637843/250000000000000 x)): 0 points increase in error, 0 points decrease in error
   (fma.f64 (Rewrite=> unpow3_binary64 (*.f64 (*.f64 x x) x)) -6450306886639899/50000000000000000 (*.f64 238732414637843/250000000000000 x)): 12 points increase in error, 6 points decrease in error
   (fma.f64 (*.f64 (*.f64 x x) x) (Rewrite<= metadata-eval (neg.f64 6450306886639899/50000000000000000)) (*.f64 238732414637843/250000000000000 x)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= fma-def_binary64 (+.f64 (*.f64 (*.f64 (*.f64 x x) x) (neg.f64 6450306886639899/50000000000000000)) (*.f64 238732414637843/250000000000000 x))): 2 points increase in error, 0 points decrease in error
   (+.f64 (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 (*.f64 (*.f64 x x) x) 6450306886639899/50000000000000000))) (*.f64 238732414637843/250000000000000 x)): 0 points increase in error, 0 points decrease in error
   (+.f64 (neg.f64 (Rewrite<= *-commutative_binary64 (*.f64 6450306886639899/50000000000000000 (*.f64 (*.f64 x x) x)))) (*.f64 238732414637843/250000000000000 x)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 238732414637843/250000000000000 x) (neg.f64 (*.f64 6450306886639899/50000000000000000 (*.f64 (*.f64 x x) x))))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= sub-neg_binary64 (-.f64 (*.f64 238732414637843/250000000000000 x) (*.f64 6450306886639899/50000000000000000 (*.f64 (*.f64 x x) x)))): 0 points increase in error, 0 points decrease in error

3. Final simplification 0.2

   \[\leadsto \mathsf{fma}\left({x}^{3}, -0.12900613773279798, x \cdot 0.954929658551372\right) \]

Alternatives

Alternative 1
Error	1.2
Cost	712

\[\begin{array}{l} t_0 := x \cdot \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\\ \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 2.75:\\ \;\;\;\;x \cdot 0.954929658551372\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.2
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7:\\ \;\;\;\;x \cdot \left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)\\ \mathbf{elif}\;x \leq 2.75:\\ \;\;\;\;x \cdot 0.954929658551372\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)\\ \end{array} \]

Alternative 3
Error	0.2
Cost	704

\[x \cdot 0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot \left(x \cdot x\right)\right) \]

Alternative 4
Error	0.2
Cost	576

\[x \cdot \left(0.954929658551372 + -0.12900613773279798 \cdot \left(x \cdot x\right)\right) \]

Alternative 5
Error	15.8
Cost	192

\[x \cdot 0.954929658551372 \]

Reproduce

herbie shell --seed 2022325 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))