fma_test1

Average Error: 61.8 → 0.4

Time: 29.5s

Precision: binary64

Cost: 1088

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

Math

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

↓

\[\frac{\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(t \cdot 6 \cdot 10^{-16}\right)}{4} + \frac{t \cdot \left(t \cdot 4 \cdot 10^{-32}\right)}{4} \]

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
 (+ (/ (* (* t 2e-16) (* t 6e-16)) 4.0) (/ (* t (* t 4e-32)) 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 (((t * 2e-16) * (t * 6e-16)) / 4.0) + ((t * (t * 4e-32)) / 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 = (((t * 2d-16) * (t * 6d-16)) / 4.0d0) + ((t * (t * 4d-32)) / 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 (((t * 2e-16) * (t * 6e-16)) / 4.0) + ((t * (t * 4e-32)) / 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 (((t * 2e-16) * (t * 6e-16)) / 4.0) + ((t * (t * 4e-32)) / 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 Float64(Float64(Float64(Float64(t * 2e-16) * Float64(t * 6e-16)) / 4.0) + Float64(Float64(t * Float64(t * 4e-32)) / 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 = (((t * 2e-16) * (t * 6e-16)) / 4.0) + ((t * (t * 4e-32)) / 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[(N[(N[(N[(t * 2e-16), $MachinePrecision] * N[(t * 6e-16), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision] + N[(N[(t * N[(t * 4e-32), $MachinePrecision]), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]

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 61.8

   \[\leadsto \color{blue}{\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - t \cdot 4 \cdot 10^{-16}\right)} \]
   Proof

   [Start] 61.8	\[ \left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \]
   rational_best-simplify-1 [=>] 61.8	\[ \left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - 2 \cdot \color{blue}{\left(2 \cdot 10^{-16} \cdot t\right)}\right) \]
   rational_best-simplify-50 [=>] 61.8	\[ \left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - \color{blue}{t \cdot \left(2 \cdot 10^{-16} \cdot 2\right)}\right) \]
   metadata-eval [=>] 61.8	\[ \left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right) + \left(-1 - t \cdot \color{blue}{4 \cdot 10^{-16}}\right) \]

3. Applied egg-rr 61.8

   \[\leadsto \color{blue}{\left(-0.5 - t \cdot 2 \cdot 10^{-16}\right) - \left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right) + \left(t \cdot 2 \cdot 10^{-16} + 0.5\right)\right)} \]

4. Simplified 57.6

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

   [Start] 61.8	\[ \left(-0.5 - t \cdot 2 \cdot 10^{-16}\right) - \left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right) + \left(t \cdot 2 \cdot 10^{-16} + 0.5\right)\right) \]
   rational_best-simplify-48 [=>] 57.6	\[ \color{blue}{\left(-0.5 - \left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right) + \left(t \cdot 2 \cdot 10^{-16} + 0.5\right)\right)\right) - t \cdot 2 \cdot 10^{-16}} \]
   rational_best-simplify-47 [=>] 57.6	\[ \left(-0.5 - \color{blue}{\left(0.5 + \left(t \cdot 2 \cdot 10^{-16} + \left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right)}\right) - t \cdot 2 \cdot 10^{-16} \]
   rational_best-simplify-57 [=>] 57.6	\[ \color{blue}{\left(\left(-0.5 - 0.5\right) + \left(-\left(t \cdot 2 \cdot 10^{-16} + \left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right)\right)} - t \cdot 2 \cdot 10^{-16} \]
   metadata-eval [=>] 57.6	\[ \left(\color{blue}{-1} + \left(-\left(t \cdot 2 \cdot 10^{-16} + \left(1 + t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right)\right) - t \cdot 2 \cdot 10^{-16} \]
   rational_best-simplify-1 [=>] 57.6	\[ \left(-1 + \left(-\left(t \cdot 2 \cdot 10^{-16} + \color{blue}{\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(1 + t \cdot 2 \cdot 10^{-16}\right)}\right)\right)\right) - t \cdot 2 \cdot 10^{-16} \]
   rational_best-simplify-3 [=>] 57.6	\[ \left(-1 + \left(-\left(t \cdot 2 \cdot 10^{-16} + \left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \color{blue}{\left(t \cdot 2 \cdot 10^{-16} + 1\right)}\right)\right)\right) - t \cdot 2 \cdot 10^{-16} \]

5. Applied egg-rr 61.8

   \[\leadsto \color{blue}{-1 + \left(\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right) + t \cdot -4 \cdot 10^{-16}\right)} \]

6. Simplified 0.5

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

   [Start] 61.8	\[ -1 + \left(\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right) + t \cdot -4 \cdot 10^{-16}\right) \]
   rational_best-simplify-3 [<=] 61.8	\[ \color{blue}{\left(\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right) + t \cdot -4 \cdot 10^{-16}\right) + -1} \]
   rational_best-simplify-19 [=>] 61.8	\[ \color{blue}{\left(\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right) + t \cdot -4 \cdot 10^{-16}\right) - 1} \]
   rational_best-simplify-59 [=>] 61.8	\[ \color{blue}{\left(t \cdot -4 \cdot 10^{-16} - \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right)} - 1 \]
   rational_best-simplify-52 [=>] 57.6	\[ \color{blue}{t \cdot -4 \cdot 10^{-16} - \left(1 + \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right)} \]
   metadata-eval [<=] 57.6	\[ t \cdot \color{blue}{\left(-2 \cdot 10^{-16} + -2 \cdot 10^{-16}\right)} - \left(1 + \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   rational_best-simplify-63 [<=] 57.6	\[ \color{blue}{\left(-2 \cdot 10^{-16} \cdot t + -2 \cdot 10^{-16} \cdot t\right)} - \left(1 + \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   rational_best-simplify-1 [<=] 57.6	\[ \left(\color{blue}{t \cdot -2 \cdot 10^{-16}} + -2 \cdot 10^{-16} \cdot t\right) - \left(1 + \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   rational_best-simplify-1 [<=] 57.6	\[ \left(t \cdot -2 \cdot 10^{-16} + \color{blue}{t \cdot -2 \cdot 10^{-16}}\right) - \left(1 + \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   rational_best-simplify-76 [=>] 61.8	\[ \color{blue}{\left(t \cdot -2 \cdot 10^{-16} - 1\right) + \left(t \cdot -2 \cdot 10^{-16} - \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right)} \]
   rational_best-simplify-18 [=>] 61.8	\[ \color{blue}{\left(t \cdot -2 \cdot 10^{-16} + -1\right)} + \left(t \cdot -2 \cdot 10^{-16} - \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   metadata-eval [<=] 61.8	\[ \left(t \cdot \color{blue}{\left(2 \cdot 10^{-16} \cdot -1\right)} + -1\right) + \left(t \cdot -2 \cdot 10^{-16} - \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   rational_best-simplify-50 [<=] 61.8	\[ \left(\color{blue}{-1 \cdot \left(2 \cdot 10^{-16} \cdot t\right)} + -1\right) + \left(t \cdot -2 \cdot 10^{-16} - \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   rational_best-simplify-1 [=>] 61.8	\[ \left(-1 \cdot \color{blue}{\left(t \cdot 2 \cdot 10^{-16}\right)} + -1\right) + \left(t \cdot -2 \cdot 10^{-16} - \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   rational_best-simplify-1 [<=] 61.8	\[ \left(\color{blue}{\left(t \cdot 2 \cdot 10^{-16}\right) \cdot -1} + -1\right) + \left(t \cdot -2 \cdot 10^{-16} - \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   rational_best-simplify-11 [<=] 61.8	\[ \left(\color{blue}{\left(-t \cdot 2 \cdot 10^{-16}\right)} + -1\right) + \left(t \cdot -2 \cdot 10^{-16} - \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   rational_best-simplify-14 [=>] 61.8	\[ \left(\color{blue}{\left(0 - t \cdot 2 \cdot 10^{-16}\right)} + -1\right) + \left(t \cdot -2 \cdot 10^{-16} - \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   metadata-eval [<=] 61.8	\[ \left(\left(0 - t \cdot 2 \cdot 10^{-16}\right) + \color{blue}{\left(-1\right)}\right) + \left(t \cdot -2 \cdot 10^{-16} - \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   rational_best-simplify-57 [<=] 61.8	\[ \color{blue}{\left(0 - \left(t \cdot 2 \cdot 10^{-16} + 1\right)\right)} + \left(t \cdot -2 \cdot 10^{-16} - \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   rational_best-simplify-21 [=>] 61.8	\[ \left(0 - \color{blue}{\left(t \cdot 2 \cdot 10^{-16} - -1\right)}\right) + \left(t \cdot -2 \cdot 10^{-16} - \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   rational_best-simplify-51 [=>] 61.8	\[ \color{blue}{\left(-1 - \left(t \cdot 2 \cdot 10^{-16} - 0\right)\right)} + \left(t \cdot -2 \cdot 10^{-16} - \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]
   rational_best-simplify-9 [=>] 61.8	\[ \left(-1 - \color{blue}{t \cdot 2 \cdot 10^{-16}}\right) + \left(t \cdot -2 \cdot 10^{-16} - \left(-\left(-1 - t \cdot 2 \cdot 10^{-16}\right) \cdot \left(-1 - t \cdot 2 \cdot 10^{-16}\right)\right)\right) \]

7. Applied egg-rr 0.4

   \[\leadsto \color{blue}{\frac{\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(t \cdot 6 \cdot 10^{-16}\right)}{4} + \frac{t \cdot \left(t \cdot 4 \cdot 10^{-32}\right)}{4}} \]

8. Final simplification 0.4

   \[\leadsto \frac{\left(t \cdot 2 \cdot 10^{-16}\right) \cdot \left(t \cdot 6 \cdot 10^{-16}\right)}{4} + \frac{t \cdot \left(t \cdot 4 \cdot 10^{-32}\right)}{4} \]

Alternatives

Alternative 1
Error	0.4
Cost	448

\[t \cdot \left(\left(t \cdot 3.2 \cdot 10^{-15}\right) \cdot 1.25 \cdot 10^{-17}\right) \]

Alternative 2
Error	0.4
Cost	320

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

Alternative 3
Error	0.4
Cost	320

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

Reproduce

herbie shell --seed 2023099 
(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)))))