fma_test1

Percentage Accurate: 3.4% → 99.4%

Time: 3.2s

Alternatives: 2

Speedup: 3.8×

Specification

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

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + t \cdot 2 \cdot 10^{-16}\\ t_1 \cdot t_1 + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (* t 2e-16))))
   (+ (* t_1 t_1) (- -1.0 (* 2.0 (* t 2e-16))))))

C

double code(double t) {
	double t_1 = 1.0 + (t * 2e-16);
	return (t_1 * t_1) + (-1.0 - (2.0 * (t * 2e-16)));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = 1.0d0 + (t * 2d-16)
    code = (t_1 * t_1) + ((-1.0d0) - (2.0d0 * (t * 2d-16)))
end function

Java

public static double code(double t) {
	double t_1 = 1.0 + (t * 2e-16);
	return (t_1 * t_1) + (-1.0 - (2.0 * (t * 2e-16)));
}

Python

def code(t):
	t_1 = 1.0 + (t * 2e-16)
	return (t_1 * t_1) + (-1.0 - (2.0 * (t * 2e-16)))

Julia

function code(t)
	t_1 = Float64(1.0 + Float64(t * 2e-16))
	return Float64(Float64(t_1 * t_1) + Float64(-1.0 - Float64(2.0 * Float64(t * 2e-16))))
end

MATLAB

function tmp = code(t)
	t_1 = 1.0 + (t * 2e-16);
	tmp = (t_1 * t_1) + (-1.0 - (2.0 * (t * 2e-16)));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(1.0 + N[(t * 2e-16), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * t$95$1), $MachinePrecision] + N[(-1.0 - N[(2.0 * N[(t * 2e-16), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 3.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + t \cdot 2 \cdot 10^{-16}\\ t_1 \cdot t_1 + \left(-1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (* t 2e-16))))
   (+ (* t_1 t_1) (- -1.0 (* 2.0 (* t 2e-16))))))

C

double code(double t) {
	double t_1 = 1.0 + (t * 2e-16);
	return (t_1 * t_1) + (-1.0 - (2.0 * (t * 2e-16)));
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    real(8) :: t_1
    t_1 = 1.0d0 + (t * 2d-16)
    code = (t_1 * t_1) + ((-1.0d0) - (2.0d0 * (t * 2d-16)))
end function

Java

public static double code(double t) {
	double t_1 = 1.0 + (t * 2e-16);
	return (t_1 * t_1) + (-1.0 - (2.0 * (t * 2e-16)));
}

Python

def code(t):
	t_1 = 1.0 + (t * 2e-16)
	return (t_1 * t_1) + (-1.0 - (2.0 * (t * 2e-16)))

Julia

function code(t)
	t_1 = Float64(1.0 + Float64(t * 2e-16))
	return Float64(Float64(t_1 * t_1) + Float64(-1.0 - Float64(2.0 * Float64(t * 2e-16))))
end

MATLAB

function tmp = code(t)
	t_1 = 1.0 + (t * 2e-16);
	tmp = (t_1 * t_1) + (-1.0 - (2.0 * (t * 2e-16)));
end

Wolfram

code[t_] := Block[{t$95$1 = N[(1.0 + N[(t * 2e-16), $MachinePrecision]), $MachinePrecision]}, N[(N[(t$95$1 * t$95$1), $MachinePrecision] + N[(-1.0 - N[(2.0 * N[(t * 2e-16), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.4% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 (* t (* t 4e-32)))

C

double code(double t) {
	return t * (t * 4e-32);
}

Fortran

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

Python

def code(t):
	return t * (t * 4e-32)

Julia

function code(t)
	return Float64(t * Float64(t * 4e-32))
end

MATLAB

function tmp = code(t)
	tmp = t * (t * 4e-32);
end

Wolfram

code[t_] := N[(t * N[(t * 4e-32), $MachinePrecision]), $MachinePrecision]

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. Step-by-step derivation

   1. associate-+r- 9.9%

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

   2. sub-neg 9.9%

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

   3. +-commutative 9.9%

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

   4. difference-of-sqr--1 9.9%

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

   5. +-commutative 9.9%

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

   6. associate--l+ 3.4%

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

   7. metadata-eval 3.4%

      \[\leadsto \left(-2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) + 1\right) \cdot \left(t \cdot 2 \cdot 10^{-16} + \color{blue}{0}\right) \]

   8. +-rgt-identity 3.4%

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

   9. distribute-lft1-in 20.8%

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

   10. *-commutative 20.8%

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

   11. associate-+r+ 20.8%

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

   12. +-commutative 20.8%

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

   13. +-commutative 20.8%

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

   14. *-rgt-identity 20.8%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 2: 99.3% accurate, 3.8× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 (* 4e-32 (* t t)))

C

double code(double t) {
	return 4e-32 * (t * t);
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = 4d-32 * (t * t)
end function

Java

public static double code(double t) {
	return 4e-32 * (t * t);
}

Python

def code(t):
	return 4e-32 * (t * t)

Julia

function code(t)
	return Float64(4e-32 * Float64(t * t))
end

MATLAB

function tmp = code(t)
	tmp = 4e-32 * (t * t);
end

Wolfram

code[t_] := N[(4e-32 * N[(t * t), $MachinePrecision]), $MachinePrecision]

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. Step-by-step derivation

   1. associate-+r- 9.9%

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

   2. sub-neg 9.9%

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

   3. +-commutative 9.9%

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

   4. difference-of-sqr--1 9.9%

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

   5. +-commutative 9.9%

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

   6. associate--l+ 3.4%

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

   7. metadata-eval 3.4%

      \[\leadsto \left(-2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) + \left(\left(1 + t \cdot 2 \cdot 10^{-16}\right) + 1\right) \cdot \left(t \cdot 2 \cdot 10^{-16} + \color{blue}{0}\right) \]

   8. +-rgt-identity 3.4%

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

   9. distribute-lft1-in 20.8%

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

   10. *-commutative 20.8%

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

   11. associate-+r+ 20.8%

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

   12. +-commutative 20.8%

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

   13. +-commutative 20.8%

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

   14. *-rgt-identity 20.8%

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

3. Simplified 99.5%

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

4. Taylor expanded in t around 0 99.4%

   \[\leadsto \color{blue}{4 \cdot 10^{-32} \cdot {t}^{2}} \]
5. Step-by-step derivation

   1. unpow2 99.4%

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

6. Simplified 99.4%

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

7. Final simplification 99.4%

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

Developer target: 21.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := 1 + t \cdot 2 \cdot 10^{-16}\\ \mathsf{fma}\left(t_1, t_1, -1 - 2 \cdot \left(t \cdot 2 \cdot 10^{-16}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (t)
 :precision binary64
 (let* ((t_1 (+ 1.0 (* t 2e-16)))) (fma t_1 t_1 (- -1.0 (* 2.0 (* t 2e-16))))))

C

double code(double t) {
	double t_1 = 1.0 + (t * 2e-16);
	return fma(t_1, t_1, (-1.0 - (2.0 * (t * 2e-16))));
}

Julia

function code(t)
	t_1 = Float64(1.0 + Float64(t * 2e-16))
	return fma(t_1, t_1, Float64(-1.0 - Float64(2.0 * Float64(t * 2e-16))))
end

Wolfram

code[t_] := Block[{t$95$1 = N[(1.0 + N[(t * 2e-16), $MachinePrecision]), $MachinePrecision]}, N[(t$95$1 * t$95$1 + N[(-1.0 - N[(2.0 * N[(t * 2e-16), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2023178 
(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)))))