fma_test1

Average Error: 61.8 → 0.4

Time: 2.4s

Precision: binary64

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

Math

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

↓

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

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 (* 2e-16 t))))

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 * (2e-16 * t));
}

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 * (2d-16 * t))
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 * (2e-16 * t));
}

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 * (2e-16 * t))

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(2e-16 * t)))
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 * (2e-16 * t));
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[(2e-16 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Error

Bits error versus t

Target

Original	61.8
Target	50.6
Herbie	0.4

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

Derivation

1. Initial program 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

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

3. Applied egg-rr 0.5

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

4. Applied egg-rr 0.4

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

5. Final simplification 0.4

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

Reproduce

herbie shell --seed 2022160 
(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)))))