Expression, p6

Percentage Accurate: 94.3% → 99.8%

Time: 14.2s

Alternatives: 7

Speedup: 1.0×

Specification

\[\left(\left(\left(-14 \leq a \land a \leq -13\right) \land \left(-3 \leq b \land b \leq -2\right)\right) \land \left(3 \leq c \land c \leq 3.5\right)\right) \land \left(12.5 \leq d \land d \leq 13.5\right)\]

Math

\[\begin{array}{l} \\ \left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (* (+ a (+ b (+ c d))) 2.0))

C

double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = (a + (b + (c + d))) * 2.0d0
end function

Java

public static double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}

Python

def code(a, b, c, d):
	return (a + (b + (c + d))) * 2.0

Julia

function code(a, b, c, d)
	return Float64(Float64(a + Float64(b + Float64(c + d))) * 2.0)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = (a + (b + (c + d))) * 2.0;
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(a + N[(b + N[(c + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Initial Program: 94.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (* (+ a (+ b (+ c d))) 2.0))

C

double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = (a + (b + (c + d))) * 2.0d0
end function

Java

public static double code(double a, double b, double c, double d) {
	return (a + (b + (c + d))) * 2.0;
}

Python

def code(a, b, c, d):
	return (a + (b + (c + d))) * 2.0

Julia

function code(a, b, c, d)
	return Float64(Float64(a + Float64(b + Float64(c + d))) * 2.0)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = (a + (b + (c + d))) * 2.0;
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(a + N[(b + N[(c + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ b \cdot \frac{2 \cdot \left(b + \left(c + \left(\mathsf{fma}\left(2, a, d\right) - a\right)\right)\right)}{b} \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (* b (/ (* 2.0 (+ b (+ c (- (fma 2.0 a d) a)))) b)))

C

double code(double a, double b, double c, double d) {
	return b * ((2.0 * (b + (c + (fma(2.0, a, d) - a)))) / b);
}

Julia

function code(a, b, c, d)
	return Float64(b * Float64(Float64(2.0 * Float64(b + Float64(c + Float64(fma(2.0, a, d) - a)))) / b))
end

Wolfram

code[a_, b_, c_, d_] := N[(b * N[(N[(2.0 * N[(b + N[(c + N[(N[(2.0 * a + d), $MachinePrecision] - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-+r+ 94.0%

      \[\leadsto \color{blue}{\left(\left(a + b\right) + \left(c + d\right)\right)} \cdot 2 \]

   2. flip-+ 93.3%

      \[\leadsto \color{blue}{\frac{\left(a + b\right) \cdot \left(a + b\right) - \left(c + d\right) \cdot \left(c + d\right)}{\left(a + b\right) - \left(c + d\right)}} \cdot 2 \]

   3. pow2 93.3%

      \[\leadsto \frac{\left(a + b\right) \cdot \left(a + b\right) - \color{blue}{{\left(c + d\right)}^{2}}}{\left(a + b\right) - \left(c + d\right)} \cdot 2 \]

4. Applied egg-rr 93.3%

   \[\leadsto \color{blue}{\frac{\left(a + b\right) \cdot \left(a + b\right) - {\left(c + d\right)}^{2}}{\left(a + b\right) - \left(c + d\right)}} \cdot 2 \]
5. Step-by-step derivation

   1. +-commutative 93.3%

      \[\leadsto \frac{\color{blue}{\left(b + a\right)} \cdot \left(a + b\right) - {\left(c + d\right)}^{2}}{\left(a + b\right) - \left(c + d\right)} \cdot 2 \]

   2. +-commutative 93.3%

      \[\leadsto \frac{\left(b + a\right) \cdot \color{blue}{\left(b + a\right)} - {\left(c + d\right)}^{2}}{\left(a + b\right) - \left(c + d\right)} \cdot 2 \]

   3. +-commutative 93.3%

      \[\leadsto \frac{\left(b + a\right) \cdot \left(b + a\right) - {\left(c + d\right)}^{2}}{\color{blue}{\left(b + a\right)} - \left(c + d\right)} \cdot 2 \]

   4. associate--l+ 93.3%

      \[\leadsto \frac{\left(b + a\right) \cdot \left(b + a\right) - {\left(c + d\right)}^{2}}{\color{blue}{b + \left(a - \left(c + d\right)\right)}} \cdot 2 \]

6. Simplified 93.3%

   \[\leadsto \color{blue}{\frac{\left(b + a\right) \cdot \left(b + a\right) - {\left(c + d\right)}^{2}}{b + \left(a - \left(c + d\right)\right)}} \cdot 2 \]

7. Taylor expanded in b around inf 94.7%

   \[\leadsto \color{blue}{b \cdot \left(2 + 2 \cdot \frac{\left(c + \left(d + 2 \cdot a\right)\right) - a}{b}\right)} \]
8. Step-by-step derivation

   1. associate--l+ 97.2%

      \[\leadsto b \cdot \left(2 + 2 \cdot \frac{\color{blue}{c + \left(\left(d + 2 \cdot a\right) - a\right)}}{b}\right) \]

   2. *-commutative 97.2%

      \[\leadsto b \cdot \left(2 + 2 \cdot \frac{c + \left(\left(d + \color{blue}{a \cdot 2}\right) - a\right)}{b}\right) \]

9. Simplified 97.2%

   \[\leadsto \color{blue}{b \cdot \left(2 + 2 \cdot \frac{c + \left(\left(d + a \cdot 2\right) - a\right)}{b}\right)} \]

10. Taylor expanded in b around 0 95.2%

    \[\leadsto b \cdot \color{blue}{\frac{2 \cdot b + 2 \cdot \left(\left(c + \left(d + 2 \cdot a\right)\right) - a\right)}{b}} \]
11. Step-by-step derivation

    1. distribute-lft-out 95.2%

       \[\leadsto b \cdot \frac{\color{blue}{2 \cdot \left(b + \left(\left(c + \left(d + 2 \cdot a\right)\right) - a\right)\right)}}{b} \]

    2. +-commutative 95.2%

       \[\leadsto b \cdot \frac{2 \cdot \left(b + \left(\left(c + \color{blue}{\left(2 \cdot a + d\right)}\right) - a\right)\right)}{b} \]

    3. fma-undefine 95.2%

       \[\leadsto b \cdot \frac{2 \cdot \left(b + \left(\left(c + \color{blue}{\mathsf{fma}\left(2, a, d\right)}\right) - a\right)\right)}{b} \]

    4. associate-+r- 99.8%

       \[\leadsto b \cdot \frac{2 \cdot \left(b + \color{blue}{\left(c + \left(\mathsf{fma}\left(2, a, d\right) - a\right)\right)}\right)}{b} \]

12. Simplified 99.8%

    \[\leadsto b \cdot \color{blue}{\frac{2 \cdot \left(b + \left(c + \left(\mathsf{fma}\left(2, a, d\right) - a\right)\right)\right)}{b}} \]
13. Add Preprocessing

Alternative 2: 97.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ b \cdot \left(2 + 2 \cdot \frac{c - \left(a - \left(d + 2 \cdot a\right)\right)}{b}\right) \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (* b (+ 2.0 (* 2.0 (/ (- c (- a (+ d (* 2.0 a)))) b)))))

C

double code(double a, double b, double c, double d) {
	return b * (2.0 + (2.0 * ((c - (a - (d + (2.0 * a)))) / b)));
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = b * (2.0d0 + (2.0d0 * ((c - (a - (d + (2.0d0 * a)))) / b)))
end function

Java

public static double code(double a, double b, double c, double d) {
	return b * (2.0 + (2.0 * ((c - (a - (d + (2.0 * a)))) / b)));
}

Python

def code(a, b, c, d):
	return b * (2.0 + (2.0 * ((c - (a - (d + (2.0 * a)))) / b)))

Julia

function code(a, b, c, d)
	return Float64(b * Float64(2.0 + Float64(2.0 * Float64(Float64(c - Float64(a - Float64(d + Float64(2.0 * a)))) / b))))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = b * (2.0 + (2.0 * ((c - (a - (d + (2.0 * a)))) / b)));
end

Wolfram

code[a_, b_, c_, d_] := N[(b * N[(2.0 + N[(2.0 * N[(N[(c - N[(a - N[(d + N[(2.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-+r+ 94.0%

      \[\leadsto \color{blue}{\left(\left(a + b\right) + \left(c + d\right)\right)} \cdot 2 \]

   2. flip-+ 93.3%

      \[\leadsto \color{blue}{\frac{\left(a + b\right) \cdot \left(a + b\right) - \left(c + d\right) \cdot \left(c + d\right)}{\left(a + b\right) - \left(c + d\right)}} \cdot 2 \]

   3. pow2 93.3%

      \[\leadsto \frac{\left(a + b\right) \cdot \left(a + b\right) - \color{blue}{{\left(c + d\right)}^{2}}}{\left(a + b\right) - \left(c + d\right)} \cdot 2 \]

4. Applied egg-rr 93.3%

   \[\leadsto \color{blue}{\frac{\left(a + b\right) \cdot \left(a + b\right) - {\left(c + d\right)}^{2}}{\left(a + b\right) - \left(c + d\right)}} \cdot 2 \]
5. Step-by-step derivation

   1. +-commutative 93.3%

      \[\leadsto \frac{\color{blue}{\left(b + a\right)} \cdot \left(a + b\right) - {\left(c + d\right)}^{2}}{\left(a + b\right) - \left(c + d\right)} \cdot 2 \]

   2. +-commutative 93.3%

      \[\leadsto \frac{\left(b + a\right) \cdot \color{blue}{\left(b + a\right)} - {\left(c + d\right)}^{2}}{\left(a + b\right) - \left(c + d\right)} \cdot 2 \]

   3. +-commutative 93.3%

      \[\leadsto \frac{\left(b + a\right) \cdot \left(b + a\right) - {\left(c + d\right)}^{2}}{\color{blue}{\left(b + a\right)} - \left(c + d\right)} \cdot 2 \]

   4. associate--l+ 93.3%

      \[\leadsto \frac{\left(b + a\right) \cdot \left(b + a\right) - {\left(c + d\right)}^{2}}{\color{blue}{b + \left(a - \left(c + d\right)\right)}} \cdot 2 \]

6. Simplified 93.3%

   \[\leadsto \color{blue}{\frac{\left(b + a\right) \cdot \left(b + a\right) - {\left(c + d\right)}^{2}}{b + \left(a - \left(c + d\right)\right)}} \cdot 2 \]

7. Taylor expanded in b around inf 94.7%

   \[\leadsto \color{blue}{b \cdot \left(2 + 2 \cdot \frac{\left(c + \left(d + 2 \cdot a\right)\right) - a}{b}\right)} \]
8. Step-by-step derivation

   1. associate--l+ 97.2%

      \[\leadsto b \cdot \left(2 + 2 \cdot \frac{\color{blue}{c + \left(\left(d + 2 \cdot a\right) - a\right)}}{b}\right) \]

   2. *-commutative 97.2%

      \[\leadsto b \cdot \left(2 + 2 \cdot \frac{c + \left(\left(d + \color{blue}{a \cdot 2}\right) - a\right)}{b}\right) \]

9. Simplified 97.2%

   \[\leadsto \color{blue}{b \cdot \left(2 + 2 \cdot \frac{c + \left(\left(d + a \cdot 2\right) - a\right)}{b}\right)} \]

10. Final simplification 97.2%

    \[\leadsto b \cdot \left(2 + 2 \cdot \frac{c - \left(a - \left(d + 2 \cdot a\right)\right)}{b}\right) \]
11. Add Preprocessing

Alternative 3: 97.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(c \cdot \left(\frac{b + \left(a + d\right)}{c} - -1\right)\right) \end{array} \]

FPCore

(FPCore (a b c d)
 :precision binary64
 (* 2.0 (* c (- (/ (+ b (+ a d)) c) -1.0))))

C

double code(double a, double b, double c, double d) {
	return 2.0 * (c * (((b + (a + d)) / c) - -1.0));
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = 2.0d0 * (c * (((b + (a + d)) / c) - (-1.0d0)))
end function

Java

public static double code(double a, double b, double c, double d) {
	return 2.0 * (c * (((b + (a + d)) / c) - -1.0));
}

Python

def code(a, b, c, d):
	return 2.0 * (c * (((b + (a + d)) / c) - -1.0))

Julia

function code(a, b, c, d)
	return Float64(2.0 * Float64(c * Float64(Float64(Float64(b + Float64(a + d)) / c) - -1.0)))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = 2.0 * (c * (((b + (a + d)) / c) - -1.0));
end

Wolfram

code[a_, b_, c_, d_] := N[(2.0 * N[(c * N[(N[(N[(b + N[(a + d), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
2. Add Preprocessing

3. Taylor expanded in c around -inf 94.9%

   \[\leadsto \color{blue}{\left(-1 \cdot \left(c \cdot \left(-1 \cdot \frac{a + \left(b + d\right)}{c} - 1\right)\right)\right)} \cdot 2 \]
4. Step-by-step derivation

   1. mul-1-neg 94.9%

      \[\leadsto \color{blue}{\left(-c \cdot \left(-1 \cdot \frac{a + \left(b + d\right)}{c} - 1\right)\right)} \cdot 2 \]

   2. *-commutative 94.9%

      \[\leadsto \left(-\color{blue}{\left(-1 \cdot \frac{a + \left(b + d\right)}{c} - 1\right) \cdot c}\right) \cdot 2 \]

   3. distribute-rgt-neg-in 94.9%

      \[\leadsto \color{blue}{\left(\left(-1 \cdot \frac{a + \left(b + d\right)}{c} - 1\right) \cdot \left(-c\right)\right)} \cdot 2 \]

   4. sub-neg 94.9%

      \[\leadsto \left(\color{blue}{\left(-1 \cdot \frac{a + \left(b + d\right)}{c} + \left(-1\right)\right)} \cdot \left(-c\right)\right) \cdot 2 \]

   5. metadata-eval 94.9%

      \[\leadsto \left(\left(-1 \cdot \frac{a + \left(b + d\right)}{c} + \color{blue}{-1}\right) \cdot \left(-c\right)\right) \cdot 2 \]

   6. +-commutative 94.9%

      \[\leadsto \left(\color{blue}{\left(-1 + -1 \cdot \frac{a + \left(b + d\right)}{c}\right)} \cdot \left(-c\right)\right) \cdot 2 \]

   7. mul-1-neg 94.9%

      \[\leadsto \left(\left(-1 + \color{blue}{\left(-\frac{a + \left(b + d\right)}{c}\right)}\right) \cdot \left(-c\right)\right) \cdot 2 \]

   8. unsub-neg 94.9%

      \[\leadsto \left(\color{blue}{\left(-1 - \frac{a + \left(b + d\right)}{c}\right)} \cdot \left(-c\right)\right) \cdot 2 \]

   9. +-commutative 94.9%

      \[\leadsto \left(\left(-1 - \frac{\color{blue}{\left(b + d\right) + a}}{c}\right) \cdot \left(-c\right)\right) \cdot 2 \]

   10. associate-+l+ 97.1%

       \[\leadsto \left(\left(-1 - \frac{\color{blue}{b + \left(d + a\right)}}{c}\right) \cdot \left(-c\right)\right) \cdot 2 \]

5. Simplified 97.1%

   \[\leadsto \color{blue}{\left(\left(-1 - \frac{b + \left(d + a\right)}{c}\right) \cdot \left(-c\right)\right)} \cdot 2 \]

6. Final simplification 97.1%

   \[\leadsto 2 \cdot \left(c \cdot \left(\frac{b + \left(a + d\right)}{c} - -1\right)\right) \]
7. Add Preprocessing

Alternative 4: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(c + \left(a + \left(b + d\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (* 2.0 (+ c (+ a (+ b d)))))

C

double code(double a, double b, double c, double d) {
	return 2.0 * (c + (a + (b + d)));
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = 2.0d0 * (c + (a + (b + d)))
end function

Java

public static double code(double a, double b, double c, double d) {
	return 2.0 * (c + (a + (b + d)));
}

Python

def code(a, b, c, d):
	return 2.0 * (c + (a + (b + d)))

Julia

function code(a, b, c, d)
	return Float64(2.0 * Float64(c + Float64(a + Float64(b + d))))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = 2.0 * (c + (a + (b + d)));
end

Wolfram

code[a_, b_, c_, d_] := N[(2.0 * N[(c + N[(a + N[(b + d), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. +-commutative 94.1%

      \[\leadsto \color{blue}{\left(\left(b + \left(c + d\right)\right) + a\right)} \cdot 2 \]

   2. +-commutative 94.1%

      \[\leadsto \left(\color{blue}{\left(\left(c + d\right) + b\right)} + a\right) \cdot 2 \]

   3. associate-+r+ 95.0%

      \[\leadsto \left(\color{blue}{\left(c + \left(d + b\right)\right)} + a\right) \cdot 2 \]

   4. +-commutative 95.0%

      \[\leadsto \left(\left(c + \color{blue}{\left(b + d\right)}\right) + a\right) \cdot 2 \]

   5. add-sqr-sqrt 93.4%

      \[\leadsto \left(\color{blue}{\sqrt{c + \left(b + d\right)} \cdot \sqrt{c + \left(b + d\right)}} + a\right) \cdot 2 \]

   6. fma-define 93.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{c + \left(b + d\right)}, \sqrt{c + \left(b + d\right)}, a\right)} \cdot 2 \]

   7. +-commutative 93.4%

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\left(b + d\right) + c}}, \sqrt{c + \left(b + d\right)}, a\right) \cdot 2 \]

   8. +-commutative 93.4%

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{\left(d + b\right)} + c}, \sqrt{c + \left(b + d\right)}, a\right) \cdot 2 \]

   9. associate-+l+ 93.8%

      \[\leadsto \mathsf{fma}\left(\sqrt{\color{blue}{d + \left(b + c\right)}}, \sqrt{c + \left(b + d\right)}, a\right) \cdot 2 \]

   10. +-commutative 93.8%

       \[\leadsto \mathsf{fma}\left(\sqrt{d + \left(b + c\right)}, \sqrt{\color{blue}{\left(b + d\right) + c}}, a\right) \cdot 2 \]

   11. +-commutative 93.8%

       \[\leadsto \mathsf{fma}\left(\sqrt{d + \left(b + c\right)}, \sqrt{\color{blue}{\left(d + b\right)} + c}, a\right) \cdot 2 \]

   12. associate-+l+ 93.6%

       \[\leadsto \mathsf{fma}\left(\sqrt{d + \left(b + c\right)}, \sqrt{\color{blue}{d + \left(b + c\right)}}, a\right) \cdot 2 \]

4. Applied egg-rr 93.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt{d + \left(b + c\right)}, \sqrt{d + \left(b + c\right)}, a\right)} \cdot 2 \]

5. Taylor expanded in c around 0 95.6%

   \[\leadsto \color{blue}{2 \cdot c + 2 \cdot \left(a + \left(b + d\right)\right)} \]
6. Step-by-step derivation

   1. distribute-lft-out 95.6%

      \[\leadsto \color{blue}{2 \cdot \left(c + \left(a + \left(b + d\right)\right)\right)} \]

7. Simplified 95.6%

   \[\leadsto \color{blue}{2 \cdot \left(c + \left(a + \left(b + d\right)\right)\right)} \]
8. Add Preprocessing

Alternative 5: 11.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ 2 \cdot \left(d + \left(c + a\right)\right) \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (* 2.0 (+ d (+ c a))))

C

double code(double a, double b, double c, double d) {
	return 2.0 * (d + (c + a));
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = 2.0d0 * (d + (c + a))
end function

Java

public static double code(double a, double b, double c, double d) {
	return 2.0 * (d + (c + a));
}

Python

def code(a, b, c, d):
	return 2.0 * (d + (c + a))

Julia

function code(a, b, c, d)
	return Float64(2.0 * Float64(d + Float64(c + a)))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = 2.0 * (d + (c + a));
end

Wolfram

code[a_, b_, c_, d_] := N[(2.0 * N[(d + N[(c + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
2. Add Preprocessing

3. Taylor expanded in b around 0 11.7%

   \[\leadsto \color{blue}{\left(a + \left(c + d\right)\right)} \cdot 2 \]
4. Step-by-step derivation

   1. +-commutative 11.7%

      \[\leadsto \color{blue}{\left(\left(c + d\right) + a\right)} \cdot 2 \]

   2. +-commutative 11.7%

      \[\leadsto \left(\color{blue}{\left(d + c\right)} + a\right) \cdot 2 \]

   3. associate-+l+ 11.7%

      \[\leadsto \color{blue}{\left(d + \left(c + a\right)\right)} \cdot 2 \]

5. Simplified 11.7%

   \[\leadsto \color{blue}{\left(d + \left(c + a\right)\right)} \cdot 2 \]

6. Final simplification 11.7%

   \[\leadsto 2 \cdot \left(d + \left(c + a\right)\right) \]
7. Add Preprocessing

Alternative 6: 11.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ 2 \cdot c \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (* 2.0 c))

C

double code(double a, double b, double c, double d) {
	return 2.0 * c;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = 2.0d0 * c
end function

Java

public static double code(double a, double b, double c, double d) {
	return 2.0 * c;
}

Python

def code(a, b, c, d):
	return 2.0 * c

Julia

function code(a, b, c, d)
	return Float64(2.0 * c)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = 2.0 * c;
end

Wolfram

code[a_, b_, c_, d_] := N[(2.0 * c), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
2. Add Preprocessing

3. Taylor expanded in c around inf 11.6%

   \[\leadsto \color{blue}{c} \cdot 2 \]

4. Final simplification 11.6%

   \[\leadsto 2 \cdot c \]
5. Add Preprocessing

Alternative 7: 6.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ b \cdot 2 \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (* b 2.0))

C

double code(double a, double b, double c, double d) {
	return b * 2.0;
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = b * 2.0d0
end function

Java

public static double code(double a, double b, double c, double d) {
	return b * 2.0;
}

Python

def code(a, b, c, d):
	return b * 2.0

Julia

function code(a, b, c, d)
	return Float64(b * 2.0)
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = b * 2.0;
end

Wolfram

code[a_, b_, c_, d_] := N[(b * 2.0), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\left(a + \left(b + \left(c + d\right)\right)\right) \cdot 2 \]
2. Add Preprocessing

3. Taylor expanded in b around inf 6.1%

   \[\leadsto \color{blue}{b} \cdot 2 \]
4. Add Preprocessing

Developer target: 94.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \left(a + b\right) \cdot 2 + \left(c + d\right) \cdot 2 \end{array} \]

FPCore

(FPCore (a b c d) :precision binary64 (+ (* (+ a b) 2.0) (* (+ c d) 2.0)))

C

double code(double a, double b, double c, double d) {
	return ((a + b) * 2.0) + ((c + d) * 2.0);
}

Fortran

real(8) function code(a, b, c, d)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: d
    code = ((a + b) * 2.0d0) + ((c + d) * 2.0d0)
end function

Java

public static double code(double a, double b, double c, double d) {
	return ((a + b) * 2.0) + ((c + d) * 2.0);
}

Python

def code(a, b, c, d):
	return ((a + b) * 2.0) + ((c + d) * 2.0)

Julia

function code(a, b, c, d)
	return Float64(Float64(Float64(a + b) * 2.0) + Float64(Float64(c + d) * 2.0))
end

MATLAB

function tmp = code(a, b, c, d)
	tmp = ((a + b) * 2.0) + ((c + d) * 2.0);
end

Wolfram

code[a_, b_, c_, d_] := N[(N[(N[(a + b), $MachinePrecision] * 2.0), $MachinePrecision] + N[(N[(c + d), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024111 
(FPCore (a b c d)
  :name "Expression, p6"
  :precision binary64
  :pre (and (and (and (and (<= -14.0 a) (<= a -13.0)) (and (<= -3.0 b) (<= b -2.0))) (and (<= 3.0 c) (<= c 3.5))) (and (<= 12.5 d) (<= d 13.5)))

  :alt
  (+ (* (+ a b) 2.0) (* (+ c d) 2.0))

  (* (+ a (+ b (+ c d))) 2.0))