fma_test1

Percentage Accurate: 3.4% → 99.3%

Time: 5.3s

Alternatives: 3

Speedup: 19.0×

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: 99.3% accurate, 3.8× speedup

Math

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

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(Float64(t * t) * 4e-32)
end

MATLAB

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

Wolfram

code[t_] := N[(N[(t * t), $MachinePrecision] * 4e-32), $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. +-commutative N/A

      \[\leadsto \left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot \left(t \cdot \frac{1}{5000000000000000} + 1\right) + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \left(\left(t \cdot \frac{1}{5000000000000000}\right) \cdot \left(1 + t \cdot \frac{1}{5000000000000000}\right) + 1 \cdot \left(1 + t \cdot \frac{1}{5000000000000000}\right)\right) + \left(\color{blue}{-1} - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   3. *-lft-identity N/A

      \[\leadsto \left(\left(t \cdot \frac{1}{5000000000000000}\right) \cdot \left(1 + t \cdot \frac{1}{5000000000000000}\right) + \left(1 + t \cdot \frac{1}{5000000000000000}\right)\right) + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \left(t \cdot \frac{1}{5000000000000000}\right) \cdot \left(1 + t \cdot \frac{1}{5000000000000000}\right) + \color{blue}{\left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot \left(t \cdot \frac{1}{5000000000000000}\right) + \left(\color{blue}{\left(1 + t \cdot \frac{1}{5000000000000000}\right)} + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) \cdot \frac{1}{5000000000000000} + \left(\color{blue}{\left(1 + t \cdot \frac{1}{5000000000000000}\right)} + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\color{blue}{\left(1 + t \cdot \frac{1}{5000000000000000}\right)} + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right)\right) \]

   8. associate-+r- N/A

      \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) + -1\right) - \color{blue}{2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) + \left(\mathsf{neg}\left(1\right)\right)\right) - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) - 1\right) - \color{blue}{2} \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(\left(t \cdot \frac{1}{5000000000000000} + 1\right) - 1\right) - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   12. *-lft-identity N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(\left(1 \cdot \left(t \cdot \frac{1}{5000000000000000}\right) + 1\right) - 1\right) - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   13. associate--l+ N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(1 \cdot \left(t \cdot \frac{1}{5000000000000000}\right) + \left(1 - 1\right)\right) - \color{blue}{2} \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(1 \cdot \left(t \cdot \frac{1}{5000000000000000}\right) + 0\right) - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   15. +-rgt-identity N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(1 \cdot \left(t \cdot \frac{1}{5000000000000000}\right) - \color{blue}{2} \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   16. *-lft-identity N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(t \cdot \frac{1}{5000000000000000} - \color{blue}{2} \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   17. associate-*r* N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(t \cdot \frac{1}{5000000000000000} - \left(2 \cdot t\right) \cdot \color{blue}{\frac{1}{5000000000000000}}\right) \]

   18. distribute-rgt-out-- N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \frac{1}{5000000000000000} \cdot \color{blue}{\left(t - 2 \cdot t\right)} \]

3. Simplified 99.4%

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

Alternative 2: 25.1% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[t_] := N[(t * 4e-32), $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. +-commutative N/A

      \[\leadsto \left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot \left(t \cdot \frac{1}{5000000000000000} + 1\right) + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \left(\left(t \cdot \frac{1}{5000000000000000}\right) \cdot \left(1 + t \cdot \frac{1}{5000000000000000}\right) + 1 \cdot \left(1 + t \cdot \frac{1}{5000000000000000}\right)\right) + \left(\color{blue}{-1} - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   3. *-lft-identity N/A

      \[\leadsto \left(\left(t \cdot \frac{1}{5000000000000000}\right) \cdot \left(1 + t \cdot \frac{1}{5000000000000000}\right) + \left(1 + t \cdot \frac{1}{5000000000000000}\right)\right) + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \left(t \cdot \frac{1}{5000000000000000}\right) \cdot \left(1 + t \cdot \frac{1}{5000000000000000}\right) + \color{blue}{\left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot \left(t \cdot \frac{1}{5000000000000000}\right) + \left(\color{blue}{\left(1 + t \cdot \frac{1}{5000000000000000}\right)} + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) \cdot \frac{1}{5000000000000000} + \left(\color{blue}{\left(1 + t \cdot \frac{1}{5000000000000000}\right)} + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\color{blue}{\left(1 + t \cdot \frac{1}{5000000000000000}\right)} + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right)\right) \]

   8. associate-+r- N/A

      \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) + -1\right) - \color{blue}{2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) + \left(\mathsf{neg}\left(1\right)\right)\right) - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) - 1\right) - \color{blue}{2} \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(\left(t \cdot \frac{1}{5000000000000000} + 1\right) - 1\right) - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   12. *-lft-identity N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(\left(1 \cdot \left(t \cdot \frac{1}{5000000000000000}\right) + 1\right) - 1\right) - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   13. associate--l+ N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(1 \cdot \left(t \cdot \frac{1}{5000000000000000}\right) + \left(1 - 1\right)\right) - \color{blue}{2} \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(1 \cdot \left(t \cdot \frac{1}{5000000000000000}\right) + 0\right) - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   15. +-rgt-identity N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(1 \cdot \left(t \cdot \frac{1}{5000000000000000}\right) - \color{blue}{2} \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   16. *-lft-identity N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(t \cdot \frac{1}{5000000000000000} - \color{blue}{2} \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   17. associate-*r* N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(t \cdot \frac{1}{5000000000000000} - \left(2 \cdot t\right) \cdot \color{blue}{\frac{1}{5000000000000000}}\right) \]

   18. distribute-rgt-out-- N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \frac{1}{5000000000000000} \cdot \color{blue}{\left(t - 2 \cdot t\right)} \]

3. Simplified 99.4%

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

   1. rem-exp-log N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{\log \left(t \cdot t\right)}\right), \frac{1}{25000000000000000000000000000000}\right) \]

   2. sum-log N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{\log t + \log t}\right), \frac{1}{25000000000000000000000000000000}\right) \]

   3. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{\frac{\log t \cdot \log t - \log t \cdot \log t}{\log t - \log t}}\right), \frac{1}{25000000000000000000000000000000}\right) \]

   4. +-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{\frac{0}{\log t - \log t}}\right), \frac{1}{25000000000000000000000000000000}\right) \]

   5. +-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{\frac{\log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}{\log t - \log t}}\right), \frac{1}{25000000000000000000000000000000}\right) \]

   6. +-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{\frac{\log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}{0}}\right), \frac{1}{25000000000000000000000000000000}\right) \]

   7. +-inverses N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{\frac{\log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) \cdot \log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}{\log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) - \log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}}\right), \frac{1}{25000000000000000000000000000000}\right) \]

   8. flip-+ N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{\log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right) + \log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}\right), \frac{1}{25000000000000000000000000000000}\right) \]

   9. log-prod N/A

      \[\leadsto \mathsf{*.f64}\left(\left(e^{\log \left({t}^{\left(\frac{\frac{2}{2}}{2}\right)} \cdot {t}^{\left(\frac{\frac{2}{2}}{2}\right)}\right)}\right), \frac{1}{25000000000000000000000000000000}\right) \]

   10. sqr-pow N/A

       \[\leadsto \mathsf{*.f64}\left(\left(e^{\log \left({t}^{\left(\frac{2}{2}\right)}\right)}\right), \frac{1}{25000000000000000000000000000000}\right) \]

   11. metadata-eval N/A

       \[\leadsto \mathsf{*.f64}\left(\left(e^{\log \left({t}^{1}\right)}\right), \frac{1}{25000000000000000000000000000000}\right) \]

   12. unpow1 N/A

       \[\leadsto \mathsf{*.f64}\left(\left(e^{\log t}\right), \frac{1}{25000000000000000000000000000000}\right) \]

   13. rem-exp-log 25.1%

       \[\leadsto \mathsf{*.f64}\left(t, \frac{1}{25000000000000000000000000000000}\right) \]

6. Applied egg-rr 25.1%

   \[\leadsto \color{blue}{t} \cdot 4 \cdot 10^{-32} \]
7. Add Preprocessing

Alternative 3: 23.6% accurate, 19.0× speedup

Math

\[\begin{array}{l} \\ 4 \cdot 10^{-32} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(t):
	return 4e-32

Julia

function code(t)
	return 4e-32
end

MATLAB

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

Wolfram

code[t_] := 4e-32

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. +-commutative N/A

      \[\leadsto \left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot \left(t \cdot \frac{1}{5000000000000000} + 1\right) + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   2. distribute-rgt-in N/A

      \[\leadsto \left(\left(t \cdot \frac{1}{5000000000000000}\right) \cdot \left(1 + t \cdot \frac{1}{5000000000000000}\right) + 1 \cdot \left(1 + t \cdot \frac{1}{5000000000000000}\right)\right) + \left(\color{blue}{-1} - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   3. *-lft-identity N/A

      \[\leadsto \left(\left(t \cdot \frac{1}{5000000000000000}\right) \cdot \left(1 + t \cdot \frac{1}{5000000000000000}\right) + \left(1 + t \cdot \frac{1}{5000000000000000}\right)\right) + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \left(t \cdot \frac{1}{5000000000000000}\right) \cdot \left(1 + t \cdot \frac{1}{5000000000000000}\right) + \color{blue}{\left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right)\right)} \]

   5. *-commutative N/A

      \[\leadsto \left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot \left(t \cdot \frac{1}{5000000000000000}\right) + \left(\color{blue}{\left(1 + t \cdot \frac{1}{5000000000000000}\right)} + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right)\right) \]

   6. associate-*r* N/A

      \[\leadsto \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) \cdot \frac{1}{5000000000000000} + \left(\color{blue}{\left(1 + t \cdot \frac{1}{5000000000000000}\right)} + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right)\right) \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\color{blue}{\left(1 + t \cdot \frac{1}{5000000000000000}\right)} + \left(-1 - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right)\right) \]

   8. associate-+r- N/A

      \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) + -1\right) - \color{blue}{2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)}\right) \]

   9. metadata-eval N/A

      \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) + \left(\mathsf{neg}\left(1\right)\right)\right) - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   10. sub-neg N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) - 1\right) - \color{blue}{2} \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(\left(t \cdot \frac{1}{5000000000000000} + 1\right) - 1\right) - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   12. *-lft-identity N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(\left(1 \cdot \left(t \cdot \frac{1}{5000000000000000}\right) + 1\right) - 1\right) - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   13. associate--l+ N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(1 \cdot \left(t \cdot \frac{1}{5000000000000000}\right) + \left(1 - 1\right)\right) - \color{blue}{2} \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(\left(1 \cdot \left(t \cdot \frac{1}{5000000000000000}\right) + 0\right) - 2 \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   15. +-rgt-identity N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(1 \cdot \left(t \cdot \frac{1}{5000000000000000}\right) - \color{blue}{2} \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   16. *-lft-identity N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(t \cdot \frac{1}{5000000000000000} - \color{blue}{2} \cdot \left(t \cdot \frac{1}{5000000000000000}\right)\right) \]

   17. associate-*r* N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \left(t \cdot \frac{1}{5000000000000000} - \left(2 \cdot t\right) \cdot \color{blue}{\frac{1}{5000000000000000}}\right) \]

   18. distribute-rgt-out-- N/A

       \[\leadsto \frac{1}{5000000000000000} \cdot \left(\left(1 + t \cdot \frac{1}{5000000000000000}\right) \cdot t\right) + \frac{1}{5000000000000000} \cdot \color{blue}{\left(t - 2 \cdot t\right)} \]

3. Simplified 99.4%

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

6. Applied egg-rr 23.5%

   \[\leadsto \color{blue}{4 \cdot 10^{-32}} \]
7. Add Preprocessing

Developer Target 1: 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 2024139 
(FPCore (t)
  :name "fma_test1"
  :precision binary64
  :pre (and (<= 0.9 t) (<= t 1.1))

  :alt
  (! :herbie-platform default (let ((x (+ 1 (* t 1/5000000000000000))) (z (- -1 (* 2 (* t 1/5000000000000000))))) (fma x x z)))

  (+ (* (+ 1.0 (* t 2e-16)) (+ 1.0 (* t 2e-16))) (- -1.0 (* 2.0 (* t 2e-16)))))