fma_test1

Percentage Accurate: 3.4% → 99.4%

Time: 7.3s

Alternatives: 2

Speedup: 3.5×

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, 2.4× speedup

Math

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

FPCore

(FPCore (t) :precision binary64 (* (* (* t 2e-16) t) 2e-16))

C

double code(double t) {
	return ((t * 2e-16) * t) * 2e-16;
}

Fortran

real(8) function code(t)
    real(8), intent (in) :: t
    code = ((t * 2d-16) * t) * 2d-16
end function

Java

public static double code(double t) {
	return ((t * 2e-16) * t) * 2e-16;
}

Python

def code(t):
	return ((t * 2e-16) * t) * 2e-16

Julia

function code(t)
	return Float64(Float64(Float64(t * 2e-16) * t) * 2e-16)
end

MATLAB

function tmp = code(t)
	tmp = ((t * 2e-16) * t) * 2e-16;
end

Wolfram

code[t_] := N[(N[(N[(t * 2e-16), $MachinePrecision] * t), $MachinePrecision] * 2e-16), $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. Add Preprocessing

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1}{25000000000000000000000000000000} \cdot {t}^{2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{{t}^{2} \cdot \frac{1}{25000000000000000000000000000000}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{{t}^{2} \cdot \frac{1}{25000000000000000000000000000000}} \]

   3. unpow2 N/A

      \[\leadsto \color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{25000000000000000000000000000000} \]

   4. lower-*.f64 99.3

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

5. Applied rewrites 99.3%

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

7. Applied rewrites 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 99.4% accurate, 3.5× speedup

Math

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

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

MATLAB

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

Wolfram

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

3. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1}{25000000000000000000000000000000} \cdot {t}^{2}} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{{t}^{2} \cdot \frac{1}{25000000000000000000000000000000}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{{t}^{2} \cdot \frac{1}{25000000000000000000000000000000}} \]

   3. unpow2 N/A

      \[\leadsto \color{blue}{\left(t \cdot t\right)} \cdot \frac{1}{25000000000000000000000000000000} \]

   4. lower-*.f64 99.3

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

5. Applied rewrites 99.3%

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

6. Taylor expanded in t around 0

   \[\leadsto \color{blue}{\frac{1}{25000000000000000000000000000000} \cdot {t}^{2}} \]
7. Step-by-step derivation

   1. unpow2 N/A

      \[\leadsto \frac{1}{25000000000000000000000000000000} \cdot \color{blue}{\left(t \cdot t\right)} \]

   2. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{25000000000000000000000000000000} \cdot t\right) \cdot t} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\frac{1}{25000000000000000000000000000000} \cdot t\right) \cdot t} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(t \cdot \frac{1}{25000000000000000000000000000000}\right)} \cdot t \]

   5. lower-*.f64 99.3

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

8. Applied rewrites 99.3%

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

9. Final simplification 99.3%

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

Developer Target 1: 21.0% accurate, 1.1× 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 2024272 
(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)))))