fma_test1

Average Error: 61.8 → 0.3

Time: 3.7s

Precision: binary64

Cost: 6848

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

Math

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

↓

\[t \cdot \sqrt{\left(t \cdot t\right) \cdot 1.6 \cdot 10^{-63}} \]

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 (* t (sqrt (* (* t t) 1.6e-63))))

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 t * sqrt(((t * t) * 1.6e-63));
}

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 = t * sqrt(((t * t) * 1.6d-63))
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 t * Math.sqrt(((t * t) * 1.6e-63));
}

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 t * math.sqrt(((t * t) * 1.6e-63))

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(t * sqrt(Float64(Float64(t * t) * 1.6e-63)))
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 = t * sqrt(((t * t) * 1.6e-63));
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[(t * N[Sqrt[N[(N[(t * t), $MachinePrecision] * 1.6e-63), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Target

Original	61.8
Target	50.6
Herbie	0.3

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

Derivation

1. Initial program 61.8

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

2. Simplified 0.4

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

   [Start] 61.8	\[ \left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   cancel-sign-sub-inv [=>] 61.8	\[ \left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \color{blue}{\left(-1 + \left(-2\right) \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)} \]
   distribute-rgt-in [=>] 61.8	\[ \color{blue}{\left(1 \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right)} + \left(-1 + \left(-2\right) \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   cancel-sign-sub-inv [<=] 61.8	\[ \left(1 \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \color{blue}{\left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)} \]
   associate-+l+ [=>] 61.8	\[ \color{blue}{1 \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\right)} \]
   *-lft-identity [=>] 61.8	\[ \color{blue}{\left(1 + t \cdot 2 \cdot 10^{-16}\right)} + \left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   +-commutative [=>] 61.8	\[ \color{blue}{\left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\right) + \left(1 + t \cdot 2 \cdot 10^{-16}\right)} \]
   associate-+r+ [<=] 62.9	\[ \color{blue}{\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(\left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right)} \]
   *-commutative [=>] 62.9	\[ \color{blue}{\left(2 \cdot 10^{-16} \cdot t\right)} \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(\left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) \]
   associate-*l* [=>] 62.9	\[ \color{blue}{2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right)} + \left(\left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) \]
   +-commutative [=>] 62.9	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \color{blue}{\left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\right)} \]
   associate-+r- [=>] 57.6	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \color{blue}{\left(\left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) + -1\right) - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)} \]
   *-lft-identity [<=] 57.6	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\left(\color{blue}{1 \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)} + -1\right) - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   +-commutative [=>] 57.6	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\left(1 \cdot \color{blue}{\left(t \cdot 2 \cdot 10^{-16} + 1\right)} + -1\right) - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   distribute-rgt-in [=>] 57.6	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\left(\color{blue}{\left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1 + 1 \cdot 1\right)} + -1\right) - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   metadata-eval [=>] 57.6	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\left(\left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1 + \color{blue}{1}\right) + -1\right) - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   associate-+l+ [=>] 50.7	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\color{blue}{\left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1 + \left(1 + -1\right)\right)} - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   metadata-eval [=>] 50.7	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1 + \color{blue}{0}\right) - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   +-rgt-identity [=>] 50.7	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\color{blue}{\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1} - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   associate-*l* [=>] 50.7	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\color{blue}{t \cdot \left(2 \cdot 10^{-16} \cdot 1\right)} - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   metadata-eval [=>] 50.7	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \left(t \cdot \color{blue}{2 \cdot 10^{-16}} - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   *-commutative [=>] 50.7	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\color{blue}{2 \cdot 10^{-16} \cdot t} - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   associate-*r* [=>] 50.7	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \left(2 \cdot 10^{-16} \cdot t - \color{blue}{\left(2 \cdot t\right) \cdot 2 \cdot 10^{-16}}\right) \]
   *-commutative [=>] 50.7	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \left(2 \cdot 10^{-16} \cdot t - \color{blue}{2 \cdot 10^{-16} \cdot \left(2 \cdot t\right)}\right) \]
   distribute-lft-out-- [=>] 50.7	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + \color{blue}{2 \cdot 10^{-16} \cdot \left(t - 2 \cdot t\right)} \]
   distribute-lft-out [=>] 49.6	\[ \color{blue}{2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(t - 2 \cdot t\right)\right)} \]
   *-lft-identity [<=] 49.6	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(\color{blue}{1 \cdot t} - 2 \cdot t\right)\right) \]
   distribute-rgt-out-- [=>] 49.6	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \color{blue}{t \cdot \left(1 - 2\right)}\right) \]
   metadata-eval [=>] 49.6	\[ 2 \cdot 10^{-16} \cdot \left(t \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + t \cdot \color{blue}{-1}\right) \]
   distribute-lft-in [<=] 49.5	\[ 2 \cdot 10^{-16} \cdot \color{blue}{\left(t \cdot \left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) + -1\right)\right)} \]
   *-rgt-identity [<=] 49.5	\[ 2 \cdot 10^{-16} \cdot \left(\color{blue}{\left(t \cdot 1\right)} \cdot \left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) + -1\right)\right) \]
   *-lft-identity [<=] 49.5	\[ 2 \cdot 10^{-16} \cdot \left(\left(t \cdot 1\right) \cdot \left(\color{blue}{1 \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)} + -1\right)\right) \]
   +-commutative [=>] 49.5	\[ 2 \cdot 10^{-16} \cdot \left(\left(t \cdot 1\right) \cdot \left(1 \cdot \color{blue}{\left(t \cdot 2 \cdot 10^{-16} + 1\right)} + -1\right)\right) \]
   distribute-rgt-in [=>] 49.5	\[ 2 \cdot 10^{-16} \cdot \left(\left(t \cdot 1\right) \cdot \left(\color{blue}{\left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1 + 1 \cdot 1\right)} + -1\right)\right) \]
   metadata-eval [=>] 49.5	\[ 2 \cdot 10^{-16} \cdot \left(\left(t \cdot 1\right) \cdot \left(\left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1 + \color{blue}{1}\right) + -1\right)\right) \]
   associate-+l+ [=>] 0.4	\[ 2 \cdot 10^{-16} \cdot \left(\left(t \cdot 1\right) \cdot \color{blue}{\left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1 + \left(1 + -1\right)\right)}\right) \]
   metadata-eval [=>] 0.4	\[ 2 \cdot 10^{-16} \cdot \left(\left(t \cdot 1\right) \cdot \left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1 + \color{blue}{0}\right)\right) \]
   +-rgt-identity [=>] 0.4	\[ 2 \cdot 10^{-16} \cdot \left(\left(t \cdot 1\right) \cdot \color{blue}{\left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1\right)}\right) \]
   associate-*l* [<=] 0.5	\[ \color{blue}{\left(2 \cdot 10^{-16} \cdot \left(t \cdot 1\right)\right) \cdot \left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1\right)} \]
   *-commutative [=>] 0.5	\[ \color{blue}{\left(\left(t \cdot 1\right) \cdot 2 \cdot 10^{-16}\right)} \cdot \left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1\right) \]
   associate-*l* [=>] 0.4	\[ \color{blue}{\left(t \cdot 1\right) \cdot \left(2 \cdot 10^{-16} \cdot \left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1\right)\right)} \]
   associate-*r* [<=] 0.4	\[ \color{blue}{t \cdot \left(1 \cdot \left(2 \cdot 10^{-16} \cdot \left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1\right)\right)\right)} \]
   *-lft-identity [=>] 0.4	\[ t \cdot \color{blue}{\left(2 \cdot 10^{-16} \cdot \left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1\right)\right)} \]
   associate-*l* [=>] 0.4	\[ t \cdot \left(2 \cdot 10^{-16} \cdot \color{blue}{\left(t \cdot \left(2 \cdot 10^{-16} \cdot 1\right)\right)}\right) \]
   metadata-eval [=>] 0.4	\[ t \cdot \left(2 \cdot 10^{-16} \cdot \left(t \cdot \color{blue}{2 \cdot 10^{-16}}\right)\right) \]
   *-commutative [=>] 0.4	\[ t \cdot \left(2 \cdot 10^{-16} \cdot \color{blue}{\left(2 \cdot 10^{-16} \cdot t\right)}\right) \]
   associate-*r* [=>] 0.6	\[ t \cdot \color{blue}{\left(\left(2 \cdot 10^{-16} \cdot 2 \cdot 10^{-16}\right) \cdot t\right)} \]
   *-commutative [<=] 0.6	\[ t \cdot \color{blue}{\left(t \cdot \left(2 \cdot 10^{-16} \cdot 2 \cdot 10^{-16}\right)\right)} \]
   metadata-eval [=>] 0.4	\[ t \cdot \left(t \cdot \color{blue}{4 \cdot 10^{-32}}\right) \]

3. Applied egg-rr 0.3

   \[\leadsto t \cdot \color{blue}{\sqrt{\left(t \cdot t\right) \cdot 1.6 \cdot 10^{-63}}} \]

4. Final simplification 0.3

   \[\leadsto t \cdot \sqrt{\left(t \cdot t\right) \cdot 1.6 \cdot 10^{-63}} \]

Alternatives

Alternative 1
Error	0.4
Cost	448

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

Alternative 2
Error	0.4
Cost	320

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

Reproduce

herbie shell --seed 2022349 
(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)))))