fma_test1

Average Accuracy: 3.4% → 99.4%

Time: 4.0s

Precision: binary64

Cost: 320

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

Math

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

↓

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

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 (* t (* t 4e-32)))

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 * (t * 4e-32);
}

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 * (t * 4d-32)
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 * (t * 4e-32);
}

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 * (t * 4e-32)

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 * Float64(t * 4e-32))
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 * (t * 4e-32);
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[(t * 4e-32), $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 99.4%

   \[\leadsto \color{blue}{t \cdot \left(t \cdot 4 \cdot 10^{-32}\right)} \]
   Step-by-step derivation

   [Start] 3.4	\[ \left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   associate-+r- [=>] 9.9	\[ \color{blue}{\left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + -1\right) - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)} \]
   sub-neg [=>] 9.9	\[ \color{blue}{\left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + -1\right) + \left(-2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)} \]
   +-commutative [=>] 9.9	\[ \color{blue}{\left(-2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + -1\right)} \]
   difference-of-sqr--1 [=>] 9.9	\[ \left(-2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \color{blue}{\left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) + 1\right) \cdot \left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) - 1\right)} \]
   +-commutative [=>] 9.9	\[ \left(-2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) + 1\right) \cdot \left(\color{blue}{\left(t \cdot 2 \cdot 10^{-16} + 1\right)} - 1\right) \]
   associate--l+ [=>] 3.4	\[ \left(-2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) + 1\right) \cdot \color{blue}{\left(t \cdot 2 \cdot 10^{-16} + \left(1 - 1\right)\right)} \]
   metadata-eval [=>] 3.4	\[ \left(-2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) + 1\right) \cdot \left(t \cdot 2 \cdot 10^{-16} + \color{blue}{0}\right) \]
   +-rgt-identity [=>] 3.4	\[ \left(-2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) + 1\right) \cdot \color{blue}{\left(t \cdot 2 \cdot 10^{-16}\right)} \]
   distribute-lft1-in [<=] 20.9	\[ \left(-2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \color{blue}{\left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(t \cdot 2 \cdot 10^{-16}\right) + t \cdot 2 \cdot 10^{-16}\right)} \]
   *-commutative [<=] 20.9	\[ \left(-2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\color{blue}{\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)} + t \cdot 2 \cdot 10^{-16}\right) \]
   associate-+r+ [=>] 20.9	\[ \color{blue}{\left(\left(-2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)\right) + t \cdot 2 \cdot 10^{-16}} \]
   +-commutative [<=] 20.9	\[ \color{blue}{\left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\right)} + t \cdot 2 \cdot 10^{-16} \]
   +-commutative [=>] 20.9	\[ \color{blue}{t \cdot 2 \cdot 10^{-16} + \left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\right)} \]
   *-rgt-identity [<=] 20.9	\[ \color{blue}{\left(t \cdot 2 \cdot 10^{-16}\right) \cdot 1} + \left(\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]

3. Final simplification 99.4%

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

Alternatives

Alternative 1
Accuracy	99.3%
Cost	320

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

Reproduce

herbie shell --seed 2023157 
(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)))))