fma_test1

Average Accuracy: 3.4% → 99.4%

Time: 4.0s

Precision: binary64

Cost: 448

\[0.9 \leq t \land t \leq 1.1\]

Math

\[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]

↓

\[2 \cdot 10^{-16} \cdot \left(t \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]

FPCore

(FPCore (t)
 :precision binary64
 (+ (* (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16))) (- -1.0 (* 2.0 (* t 2e-16)))))

↓

(FPCore (t) :precision binary64 (* 2e-16 (* t (* t 2e-16))))

C

double code(double t) {
	return ((1.0 + (t * 2e-16)) * (1.0 + (t * 2e-16))) + (-1.0 - (2.0 * (t * 2e-16)));
}

↓

double code(double t) {
	return 2e-16 * (t * (t * 2e-16));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = ((1.0d0 + (t * 2d-16)) * (1.0d0 + (t * 2d-16))) + ((-1.0d0) - (2.0d0 * (t * 2d-16)))
end function

↓

real(8) function code(t)
    real(8), intent (in) :: t
    code = 2d-16 * (t * (t * 2d-16))
end function

Java

public static double code(double t) {
	return ((1.0 + (t * 2e-16)) * (1.0 + (t * 2e-16))) + (-1.0 - (2.0 * (t * 2e-16)));
}

↓

public static double code(double t) {
	return 2e-16 * (t * (t * 2e-16));
}

Python

def code(t):
	return ((1.0 + (t * 2e-16)) * (1.0 + (t * 2e-16))) + (-1.0 - (2.0 * (t * 2e-16)))

↓

def code(t):
	return 2e-16 * (t * (t * 2e-16))

Julia

function code(t)
	return Float64(Float64(Float64(1.0 + Float64(t * 2e-16)) * Float64(1.0 + Float64(t * 2e-16))) + Float64(-1.0 - Float64(2.0 * Float64(t * 2e-16))))
end

↓

function code(t)
	return Float64(2e-16 * Float64(t * Float64(t * 2e-16)))
end

MATLAB

function tmp = code(t)
	tmp = ((1.0 + (t * 2e-16)) * (1.0 + (t * 2e-16))) + (-1.0 - (2.0 * (t * 2e-16)));
end

↓

function tmp = code(t)
	tmp = 2e-16 * (t * (t * 2e-16));
end

Wolfram

code[t_] := N[(N[(N[(1.0 + N[(t * 2e-16), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(t * 2e-16), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.0 - N[(2.0 * N[(t * 2e-16), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[t_] := N[(2e-16 * N[(t * N[(t * 2e-16), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	3.4%
Target	21.0%
Herbie	99.4%

\[\mathsf{fma}\left(1 + t \cdot 2 \cdot 10^{-16}, 1 + t \cdot 2 \cdot 10^{-16}, -1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]

Derivation

1. Initial program 3.4%

   \[\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]

2. Simplified 99.3%

   \[\leadsto \color{blue}{t \cdot \left(t \cdot 4 \cdot 10^{-32}\right)} \]
   Proof

   [Start] 3.4	\[ \left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   cancel-sign-sub-inv [=>] 3.4	\[ \left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \color{blue}{\left(-1 + \left(-2\right) \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)} \]
   distribute-rgt-in [=>] 3.4	\[ \color{blue}{\left(1 \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right)} + \left(-1 + \left(-2\right) \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   cancel-sign-sub-inv [<=] 3.4	\[ \left(1 \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \color{blue}{\left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)} \]
   associate-+l+ [=>] 3.4	\[ \color{blue}{1 \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\right)} \]
   *-lft-identity [=>] 3.4	\[ \color{blue}{\left(1 + t \cdot 2 \cdot 10^{-16}\right)} + \left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   +-commutative [=>] 3.4	\[ \color{blue}{\left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\right) + \left(1 + t \cdot 2 \cdot 10^{-16}\right)} \]
   associate-+r+ [<=] 1.7	\[ \color{blue}{\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(\left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right)} \]
   *-commutative [=>] 1.7	\[ \color{blue}{\left(2 \cdot 10^{-16} \cdot t\right)} \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(\left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) \]
   associate-*l* [=>] 1.7	\[ \color{blue}{2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right)} + \left(\left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) \]
   +-commutative [=>] 1.7	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \color{blue}{\left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\right)} \]
   associate-+r- [=>] 10.0	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \color{blue}{\left(\left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) + -1\right) - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)} \]

3. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{{\left(t \cdot 2 \cdot 10^{-16}\right)}^{2}} \]
   Proof

   [Start] 99.3	\[ t \cdot \left(t \cdot 4 \cdot 10^{-32}\right) \]
   add-sqr-sqrt [=>] 99.0	\[ \color{blue}{\sqrt{t \cdot \left(t \cdot 4 \cdot 10^{-32}\right)} \cdot \sqrt{t \cdot \left(t \cdot 4 \cdot 10^{-32}\right)}} \]
   pow2 [=>] 99.0	\[ \color{blue}{{\left(\sqrt{t \cdot \left(t \cdot 4 \cdot 10^{-32}\right)}\right)}^{2}} \]
   associate-*r* [=>] 99.0	\[ {\left(\sqrt{\color{blue}{\left(t \cdot t\right) \cdot 4 \cdot 10^{-32}}}\right)}^{2} \]
   sqrt-prod [=>] 99.2	\[ {\color{blue}{\left(\sqrt{t \cdot t} \cdot \sqrt{4 \cdot 10^{-32}}\right)}}^{2} \]
   sqrt-unprod [<=] 98.4	\[ {\left(\color{blue}{\left(\sqrt{t} \cdot \sqrt{t}\right)} \cdot \sqrt{4 \cdot 10^{-32}}\right)}^{2} \]
   add-sqr-sqrt [<=] 99.2	\[ {\left(\color{blue}{t} \cdot \sqrt{4 \cdot 10^{-32}}\right)}^{2} \]
   metadata-eval [=>] 99.2	\[ {\left(t \cdot \color{blue}{2 \cdot 10^{-16}}\right)}^{2} \]

4. Applied egg-rr 99.4%

   \[\leadsto \color{blue}{\left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot t\right) \cdot 2 \cdot 10^{-16}} \]
   Proof

   [Start] 99.2	\[ {\left(t \cdot 2 \cdot 10^{-16}\right)}^{2} \]
   unpow2 [=>] 99.2	\[ \color{blue}{\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(t \cdot 2 \cdot 10^{-16}\right)} \]
   associate-*r* [=>] 99.4	\[ \color{blue}{\left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot t\right) \cdot 2 \cdot 10^{-16}} \]

5. Final simplification 99.4%

   \[\leadsto 2 \cdot 10^{-16} \cdot \left(t \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]

Alternatives

Alternative 1
Accuracy	99.4%
Cost	448

\[t \cdot \left(2 \cdot 10^{-16} \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]

Alternative 2
Accuracy	99.3%
Cost	320

\[4 \cdot 10^{-32} \cdot \left(t \cdot t\right) \]

Reproduce

herbie shell --seed 2023126 
(FPCore (t)
  :name "fma_test1"
  :precision binary64
  :pre (and (<= 0.9 t) (<= t 1.1))

  :herbie-target
  (fma (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16)) (- -1.0 (* 2.0 (* t 2e-16))))

  (+ (* (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16))) (- -1.0 (* 2.0 (* t 2e-16)))))