fma_test1

Average Error: 61.8 → 0.2

Time: 3.8s

Precision: binary64

Cost: 13056

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

Math

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

↓

\[\sqrt{1.6 \cdot 10^{-63} \cdot {t}^{4}} \]

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 (sqrt (* 1.6e-63 (pow t 4.0))))

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 sqrt((1.6e-63 * pow(t, 4.0)));
}

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 = sqrt((1.6d-63 * (t ** 4.0d0)))
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 Math.sqrt((1.6e-63 * Math.pow(t, 4.0)));
}

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 math.sqrt((1.6e-63 * math.pow(t, 4.0)))

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 sqrt(Float64(1.6e-63 * (t ^ 4.0)))
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 = sqrt((1.6e-63 * (t ^ 4.0)));
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[Sqrt[N[(1.6e-63 * N[Power[t, 4.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Target

Original	61.8
Target	50.6
Herbie	0.2

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

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

3. Taylor expanded in t around 0 0.4

   \[\leadsto \color{blue}{4 \cdot 10^{-32} \cdot {t}^{2}} \]

4. Simplified 0.4

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

   [Start] 0.4	\[ 4 \cdot 10^{-32} \cdot {t}^{2} \]
   unpow2 [=>] 0.4	\[ 4 \cdot 10^{-32} \cdot \color{blue}{\left(t \cdot t\right)} \]

5. Applied egg-rr 0.2

   \[\leadsto \color{blue}{\sqrt{1.6 \cdot 10^{-63} \cdot {t}^{4}}} \]

6. Final simplification 0.2

   \[\leadsto \sqrt{1.6 \cdot 10^{-63} \cdot {t}^{4}} \]

Alternatives

Alternative 1
Error	0.3
Cost	6848

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

Alternative 2
Error	0.4
Cost	448

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

Alternative 3
Error	0.4
Cost	448

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

Alternative 4
Error	0.4
Cost	320

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

Reproduce

herbie shell --seed 2023066 
(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)))))