exp2 (problem 3.3.7)

Percentage Accurate: 53.6% → 99.5%

Time: 10.0s

Alternatives: 8

Speedup: 68.7×

Specification

\[\left|x\right| \leq 710\]

Math

\[\begin{array}{l} \\ \left(e^{x} - 2\right) + e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (- (exp x) 2.0) (exp (- x))))

C

double code(double x) {
	return (exp(x) - 2.0) + exp(-x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - 2.0d0) + exp(-x)
end function

Java

public static double code(double x) {
	return (Math.exp(x) - 2.0) + Math.exp(-x);
}

Python

def code(x):
	return (math.exp(x) - 2.0) + math.exp(-x)

Julia

function code(x)
	return Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - 2.0) + exp(-x);
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Initial Program: 53.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(e^{x} - 2\right) + e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ (- (exp x) 2.0) (exp (- x))))

C

double code(double x) {
	return (exp(x) - 2.0) + exp(-x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - 2.0d0) + exp(-x)
end function

Java

public static double code(double x) {
	return (Math.exp(x) - 2.0) + Math.exp(-x);
}

Python

def code(x):
	return (math.exp(x) - 2.0) + math.exp(-x)

Julia

function code(x)
	return Float64(Float64(exp(x) - 2.0) + exp(Float64(-x)))
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - 2.0) + exp(-x);
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000145:\\ \;\;\;\;x\_m \cdot x\_m\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \cosh x\_m - 2\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000145) (* x_m x_m) (- (* 2.0 (cosh x_m)) 2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.000145) {
		tmp = x_m * x_m;
	} else {
		tmp = (2.0 * cosh(x_m)) - 2.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.000145d0) then
        tmp = x_m * x_m
    else
        tmp = (2.0d0 * cosh(x_m)) - 2.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.000145) {
		tmp = x_m * x_m;
	} else {
		tmp = (2.0 * Math.cosh(x_m)) - 2.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.000145:
		tmp = x_m * x_m
	else:
		tmp = (2.0 * math.cosh(x_m)) - 2.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.000145)
		tmp = Float64(x_m * x_m);
	else
		tmp = Float64(Float64(2.0 * cosh(x_m)) - 2.0);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.000145)
		tmp = x_m * x_m;
	else
		tmp = (2.0 * cosh(x_m)) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.000145], N[(x$95$m * x$95$m), $MachinePrecision], N[(N[(2.0 * N[Cosh[x$95$m], $MachinePrecision]), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.45e-4

   1. Initial program 56.1%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 98.9%

      \[\leadsto \color{blue}{{x}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 99.4%

         \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4}\right) \]

   5. Applied egg-rr 98.9%

      \[\leadsto \color{blue}{x \cdot x} \]

   if 1.45e-4 < x

   1. Initial program 94.6%

      \[\left(e^{x} - 2\right) + e^{-x} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. +-commutative 94.6%

         \[\leadsto \color{blue}{e^{-x} + \left(e^{x} - 2\right)} \]

      2. associate-+r- 95.4%

         \[\leadsto \color{blue}{\left(e^{-x} + e^{x}\right) - 2} \]

      3. add-sqr-sqrt 0.0%

         \[\leadsto \left(e^{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}} + e^{x}\right) - 2 \]

      4. sqrt-unprod 16.6%

         \[\leadsto \left(e^{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}} + e^{x}\right) - 2 \]

      5. sqr-neg 16.6%

         \[\leadsto \left(e^{\sqrt{\color{blue}{x \cdot x}}} + e^{x}\right) - 2 \]

      6. sqrt-unprod 16.6%

         \[\leadsto \left(e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} + e^{x}\right) - 2 \]

      7. add-sqr-sqrt 16.6%

         \[\leadsto \left(e^{\color{blue}{x}} + e^{x}\right) - 2 \]

      8. add-sqr-sqrt 16.6%

         \[\leadsto \left(e^{x} + e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}\right) - 2 \]

      9. sqrt-unprod 16.6%

         \[\leadsto \left(e^{x} + e^{\color{blue}{\sqrt{x \cdot x}}}\right) - 2 \]

      10. sqr-neg 16.6%

          \[\leadsto \left(e^{x} + e^{\sqrt{\color{blue}{\left(-x\right) \cdot \left(-x\right)}}}\right) - 2 \]

      11. sqrt-unprod 0.0%

          \[\leadsto \left(e^{x} + e^{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}\right) - 2 \]

      12. add-sqr-sqrt 95.4%

          \[\leadsto \left(e^{x} + e^{\color{blue}{-x}}\right) - 2 \]

      13. cosh-undef 95.4%

          \[\leadsto \color{blue}{2 \cdot \cosh x} - 2 \]

   4. Applied egg-rr 95.4%

      \[\leadsto \color{blue}{2 \cdot \cosh x - 2} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \mathsf{fma}\left(x\_m, x\_m, \mathsf{fma}\left(x\_m \cdot x\_m, 0.002777777777777778, 0.08333333333333333\right) \cdot {x\_m}^{4}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (fma
  x_m
  x_m
  (*
   (fma (* x_m x_m) 0.002777777777777778 0.08333333333333333)
   (pow x_m 4.0))))

C

x_m = fabs(x);
double code(double x_m) {
	return fma(x_m, x_m, (fma((x_m * x_m), 0.002777777777777778, 0.08333333333333333) * pow(x_m, 4.0)));
}

Julia

x_m = abs(x)
function code(x_m)
	return fma(x_m, x_m, Float64(fma(Float64(x_m * x_m), 0.002777777777777778, 0.08333333333333333) * (x_m ^ 4.0)))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * x$95$m + N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision] * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.8%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative 98.8%

      \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + \color{blue}{{x}^{2} \cdot 0.002777777777777778}\right)\right) \]

5. Simplified 98.8%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in 98.8%

      \[\leadsto \color{blue}{1 \cdot {x}^{2} + \left({x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right) \cdot {x}^{2}} \]

   2. *-un-lft-identity 98.8%

      \[\leadsto \color{blue}{{x}^{2}} + \left({x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right) \cdot {x}^{2} \]

   3. unpow2 98.8%

      \[\leadsto \color{blue}{x \cdot x} + \left({x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right) \cdot {x}^{2} \]

   4. fma-define 98.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \left({x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right) \cdot {x}^{2}\right)} \]

   5. *-commutative 98.8%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(\left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right) \cdot {x}^{2}\right)} \cdot {x}^{2}\right) \]

   6. associate-*l* 98.8%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right) \cdot \left({x}^{2} \cdot {x}^{2}\right)}\right) \]

   7. +-commutative 98.8%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\left({x}^{2} \cdot 0.002777777777777778 + 0.08333333333333333\right)} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \]

   8. fma-define 98.8%

      \[\leadsto \mathsf{fma}\left(x, x, \color{blue}{\mathsf{fma}\left({x}^{2}, 0.002777777777777778, 0.08333333333333333\right)} \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \]

   9. pow-prod-up 98.8%

      \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left({x}^{2}, 0.002777777777777778, 0.08333333333333333\right) \cdot \color{blue}{{x}^{\left(2 + 2\right)}}\right) \]

   10. metadata-eval 98.8%

       \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left({x}^{2}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{\color{blue}{4}}\right) \]

7. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, \mathsf{fma}\left({x}^{2}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4}\right)} \]
8. Step-by-step derivation

   1. unpow2 98.8%

      \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4}\right) \]

9. Applied egg-rr 98.8%

   \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4}\right) \]
10. Add Preprocessing

Alternative 3: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {x\_m}^{2} \cdot \left(1 + {x\_m}^{2} \cdot \left(0.08333333333333333 + \left(x\_m \cdot x\_m\right) \cdot 0.002777777777777778\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow x_m 2.0)
  (+
   1.0
   (*
    (pow x_m 2.0)
    (+ 0.08333333333333333 (* (* x_m x_m) 0.002777777777777778))))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(x_m, 2.0) * (1.0 + (pow(x_m, 2.0) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (x_m ** 2.0d0) * (1.0d0 + ((x_m ** 2.0d0) * (0.08333333333333333d0 + ((x_m * x_m) * 0.002777777777777778d0))))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(x_m, 2.0) * (1.0 + (Math.pow(x_m, 2.0) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(x_m, 2.0) * (1.0 + (math.pow(x_m, 2.0) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((x_m ^ 2.0) * Float64(1.0 + Float64((x_m ^ 2.0) * Float64(0.08333333333333333 + Float64(Float64(x_m * x_m) * 0.002777777777777778)))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m ^ 2.0) * (1.0 + ((x_m ^ 2.0) * (0.08333333333333333 + ((x_m * x_m) * 0.002777777777777778))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(1.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.08333333333333333 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.002777777777777778), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.8%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + 0.002777777777777778 \cdot {x}^{2}\right)\right)} \]
4. Step-by-step derivation

   1. *-commutative 98.8%

      \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + \color{blue}{{x}^{2} \cdot 0.002777777777777778}\right)\right) \]

5. Simplified 98.8%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + {x}^{2} \cdot 0.002777777777777778\right)\right)} \]
6. Step-by-step derivation

   1. unpow2 98.8%

      \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4}\right) \]

7. Applied egg-rr 98.8%

   \[\leadsto {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(0.08333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.002777777777777778\right)\right) \]
8. Add Preprocessing

Alternative 4: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \mathsf{fma}\left(x\_m, x\_m, 0.08333333333333333 \cdot {x\_m}^{4}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (fma x_m x_m (* 0.08333333333333333 (pow x_m 4.0))))

C

x_m = fabs(x);
double code(double x_m) {
	return fma(x_m, x_m, (0.08333333333333333 * pow(x_m, 4.0)));
}

Julia

x_m = abs(x)
function code(x_m)
	return fma(x_m, x_m, Float64(0.08333333333333333 * (x_m ^ 4.0)))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * x$95$m + N[(0.08333333333333333 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.5%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-in 98.5%

      \[\leadsto \color{blue}{1 \cdot {x}^{2} + \left(0.08333333333333333 \cdot {x}^{2}\right) \cdot {x}^{2}} \]

   2. *-lft-identity 98.5%

      \[\leadsto \color{blue}{{x}^{2}} + \left(0.08333333333333333 \cdot {x}^{2}\right) \cdot {x}^{2} \]

   3. associate-*l* 98.5%

      \[\leadsto {x}^{2} + \color{blue}{0.08333333333333333 \cdot \left({x}^{2} \cdot {x}^{2}\right)} \]

   4. pow-sqr 98.5%

      \[\leadsto {x}^{2} + 0.08333333333333333 \cdot \color{blue}{{x}^{\left(2 \cdot 2\right)}} \]

   5. metadata-eval 98.5%

      \[\leadsto {x}^{2} + 0.08333333333333333 \cdot {x}^{\color{blue}{4}} \]

5. Simplified 98.5%

   \[\leadsto \color{blue}{{x}^{2} + 0.08333333333333333 \cdot {x}^{4}} \]
6. Step-by-step derivation

   1. unpow2 98.5%

      \[\leadsto \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4} \]

   2. fma-define 98.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)} \]

7. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)} \]
8. Add Preprocessing

Alternative 5: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {x\_m}^{2} \cdot \left(1 + \left(x\_m \cdot x\_m\right) \cdot 0.08333333333333333\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (pow x_m 2.0) (+ 1.0 (* (* x_m x_m) 0.08333333333333333))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(x_m, 2.0) * (1.0 + ((x_m * x_m) * 0.08333333333333333));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (x_m ** 2.0d0) * (1.0d0 + ((x_m * x_m) * 0.08333333333333333d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(x_m, 2.0) * (1.0 + ((x_m * x_m) * 0.08333333333333333));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(x_m, 2.0) * (1.0 + ((x_m * x_m) * 0.08333333333333333))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((x_m ^ 2.0) * Float64(1.0 + Float64(Float64(x_m * x_m) * 0.08333333333333333)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m ^ 2.0) * (1.0 + ((x_m * x_m) * 0.08333333333333333));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(1.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.5%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. *-commutative 98.5%

      \[\leadsto {x}^{2} \cdot \left(1 + \color{blue}{{x}^{2} \cdot 0.08333333333333333}\right) \]

5. Simplified 98.5%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot 0.08333333333333333\right)} \]
6. Step-by-step derivation

   1. unpow2 98.8%

      \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4}\right) \]

7. Applied egg-rr 98.5%

   \[\leadsto {x}^{2} \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.08333333333333333\right) \]
8. Add Preprocessing

Alternative 6: 99.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot x\_m + 0.08333333333333333 \cdot {x\_m}^{4} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (+ (* x_m x_m) (* 0.08333333333333333 (pow x_m 4.0))))

C

x_m = fabs(x);
double code(double x_m) {
	return (x_m * x_m) + (0.08333333333333333 * pow(x_m, 4.0));
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = (x_m * x_m) + (0.08333333333333333d0 * (x_m ** 4.0d0))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * x_m) + (0.08333333333333333 * Math.pow(x_m, 4.0));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * x_m) + (0.08333333333333333 * math.pow(x_m, 4.0))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(x_m * x_m) + Float64(0.08333333333333333 * (x_m ^ 4.0)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * x_m) + (0.08333333333333333 * (x_m ^ 4.0));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * x$95$m), $MachinePrecision] + N[(0.08333333333333333 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.5%

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + 0.08333333333333333 \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. distribute-rgt-in 98.5%

      \[\leadsto \color{blue}{1 \cdot {x}^{2} + \left(0.08333333333333333 \cdot {x}^{2}\right) \cdot {x}^{2}} \]

   2. *-lft-identity 98.5%

      \[\leadsto \color{blue}{{x}^{2}} + \left(0.08333333333333333 \cdot {x}^{2}\right) \cdot {x}^{2} \]

   3. associate-*l* 98.5%

      \[\leadsto {x}^{2} + \color{blue}{0.08333333333333333 \cdot \left({x}^{2} \cdot {x}^{2}\right)} \]

   4. pow-sqr 98.5%

      \[\leadsto {x}^{2} + 0.08333333333333333 \cdot \color{blue}{{x}^{\left(2 \cdot 2\right)}} \]

   5. metadata-eval 98.5%

      \[\leadsto {x}^{2} + 0.08333333333333333 \cdot {x}^{\color{blue}{4}} \]

5. Simplified 98.5%

   \[\leadsto \color{blue}{{x}^{2} + 0.08333333333333333 \cdot {x}^{4}} \]
6. Step-by-step derivation

   1. unpow2 98.8%

      \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4}\right) \]

7. Applied egg-rr 98.5%

   \[\leadsto \color{blue}{x \cdot x} + 0.08333333333333333 \cdot {x}^{4} \]
8. Add Preprocessing

Alternative 7: 98.5% accurate, 68.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot x\_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* x_m x_m))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * x_m;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = x_m * x_m
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * x_m;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * x_m

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * x_m)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m * x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * x$95$m), $MachinePrecision]

Derivation

1. Initial program 56.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 98.0%

   \[\leadsto \color{blue}{{x}^{2}} \]
4. Step-by-step derivation

   1. unpow2 98.8%

      \[\leadsto \mathsf{fma}\left(x, x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.002777777777777778, 0.08333333333333333\right) \cdot {x}^{4}\right) \]

5. Applied egg-rr 98.0%

   \[\leadsto \color{blue}{x \cdot x} \]
6. Add Preprocessing

Alternative 8: 4.3% accurate, 206.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 2 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 2.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 2.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 2.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 2.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 2.0

Julia

x_m = abs(x)
function code(x_m)
	return 2.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 2.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 2.0

Derivation

1. Initial program 56.5%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. flip-- 56.2%

      \[\leadsto \color{blue}{\frac{e^{x} \cdot e^{x} - 2 \cdot 2}{e^{x} + 2}} + e^{-x} \]

   2. clear-num 56.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} + 2}{e^{x} \cdot e^{x} - 2 \cdot 2}}} + e^{-x} \]

   3. sub-neg 56.1%

      \[\leadsto \frac{1}{\frac{e^{x} + 2}{\color{blue}{e^{x} \cdot e^{x} + \left(-2 \cdot 2\right)}}} + e^{-x} \]

   4. pow2 56.1%

      \[\leadsto \frac{1}{\frac{e^{x} + 2}{\color{blue}{{\left(e^{x}\right)}^{2}} + \left(-2 \cdot 2\right)}} + e^{-x} \]

   5. metadata-eval 56.1%

      \[\leadsto \frac{1}{\frac{e^{x} + 2}{{\left(e^{x}\right)}^{2} + \left(-\color{blue}{4}\right)}} + e^{-x} \]

   6. metadata-eval 56.1%

      \[\leadsto \frac{1}{\frac{e^{x} + 2}{{\left(e^{x}\right)}^{2} + \color{blue}{-4}}} + e^{-x} \]

4. Applied egg-rr 56.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} + 2}{{\left(e^{x}\right)}^{2} + -4}}} + e^{-x} \]
5. Step-by-step derivation

   1. frac-2neg 56.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{-\left(e^{x} + 2\right)}{-\left({\left(e^{x}\right)}^{2} + -4\right)}}} + e^{-x} \]

   2. distribute-frac-neg 56.1%

      \[\leadsto \frac{1}{\color{blue}{-\frac{e^{x} + 2}{-\left({\left(e^{x}\right)}^{2} + -4\right)}}} + e^{-x} \]

   3. distribute-neg-frac2 56.1%

      \[\leadsto \frac{1}{-\color{blue}{\left(-\frac{e^{x} + 2}{{\left(e^{x}\right)}^{2} + -4}\right)}} + e^{-x} \]

   4. /-rgt-identity 56.1%

      \[\leadsto \frac{1}{-\left(-\color{blue}{\frac{\frac{e^{x} + 2}{{\left(e^{x}\right)}^{2} + -4}}{1}}\right)} + e^{-x} \]

   5. clear-num 56.1%

      \[\leadsto \frac{1}{-\left(-\color{blue}{\frac{1}{\frac{1}{\frac{e^{x} + 2}{{\left(e^{x}\right)}^{2} + -4}}}}\right)} + e^{-x} \]

   6. distribute-neg-frac 56.1%

      \[\leadsto \frac{1}{-\color{blue}{\frac{-1}{\frac{1}{\frac{e^{x} + 2}{{\left(e^{x}\right)}^{2} + -4}}}}} + e^{-x} \]

   7. metadata-eval 56.1%

      \[\leadsto \frac{1}{-\frac{\color{blue}{-1}}{\frac{1}{\frac{e^{x} + 2}{{\left(e^{x}\right)}^{2} + -4}}}} + e^{-x} \]

   8. clear-num 56.2%

      \[\leadsto \frac{1}{-\frac{-1}{\color{blue}{\frac{{\left(e^{x}\right)}^{2} + -4}{e^{x} + 2}}}} + e^{-x} \]

   9. metadata-eval 56.2%

      \[\leadsto \frac{1}{-\frac{-1}{\frac{{\left(e^{x}\right)}^{2} + \color{blue}{\left(-4\right)}}{e^{x} + 2}}} + e^{-x} \]

   10. sub-neg 56.2%

       \[\leadsto \frac{1}{-\frac{-1}{\frac{\color{blue}{{\left(e^{x}\right)}^{2} - 4}}{e^{x} + 2}}} + e^{-x} \]

   11. unpow2 56.2%

       \[\leadsto \frac{1}{-\frac{-1}{\frac{\color{blue}{e^{x} \cdot e^{x}} - 4}{e^{x} + 2}}} + e^{-x} \]

   12. metadata-eval 56.2%

       \[\leadsto \frac{1}{-\frac{-1}{\frac{e^{x} \cdot e^{x} - \color{blue}{2 \cdot 2}}{e^{x} + 2}}} + e^{-x} \]

   13. flip-- 56.5%

       \[\leadsto \frac{1}{-\frac{-1}{\color{blue}{e^{x} - 2}}} + e^{-x} \]

   14. sub-neg 56.5%

       \[\leadsto \frac{1}{-\frac{-1}{\color{blue}{e^{x} + \left(-2\right)}}} + e^{-x} \]

   15. metadata-eval 56.5%

       \[\leadsto \frac{1}{-\frac{-1}{e^{x} + \color{blue}{-2}}} + e^{-x} \]

6. Applied egg-rr 56.5%

   \[\leadsto \frac{1}{\color{blue}{-\frac{-1}{e^{x} + -2}}} + e^{-x} \]
7. Step-by-step derivation

   1. +-commutative 56.5%

      \[\leadsto \color{blue}{e^{-x} + \frac{1}{-\frac{-1}{e^{x} + -2}}} \]

   2. distribute-frac-neg2 56.5%

      \[\leadsto e^{-x} + \color{blue}{\left(-\frac{1}{\frac{-1}{e^{x} + -2}}\right)} \]

   3. clear-num 56.5%

      \[\leadsto e^{-x} + \left(-\color{blue}{\frac{e^{x} + -2}{-1}}\right) \]

   4. unsub-neg 56.5%

      \[\leadsto \color{blue}{e^{-x} - \frac{e^{x} + -2}{-1}} \]

   5. clear-num 56.5%

      \[\leadsto e^{-x} - \color{blue}{\frac{1}{\frac{-1}{e^{x} + -2}}} \]

   6. add-sqr-sqrt 55.6%

      \[\leadsto e^{-x} - \frac{1}{\color{blue}{\sqrt{\frac{-1}{e^{x} + -2}} \cdot \sqrt{\frac{-1}{e^{x} + -2}}}} \]

   7. sqrt-unprod 55.8%

      \[\leadsto e^{-x} - \frac{1}{\color{blue}{\sqrt{\frac{-1}{e^{x} + -2} \cdot \frac{-1}{e^{x} + -2}}}} \]

   8. sqr-neg 55.8%

      \[\leadsto e^{-x} - \frac{1}{\sqrt{\color{blue}{\left(-\frac{-1}{e^{x} + -2}\right) \cdot \left(-\frac{-1}{e^{x} + -2}\right)}}} \]

   9. sqrt-unprod 0.1%

      \[\leadsto e^{-x} - \frac{1}{\color{blue}{\sqrt{-\frac{-1}{e^{x} + -2}} \cdot \sqrt{-\frac{-1}{e^{x} + -2}}}} \]

   10. add-sqr-sqrt 4.6%

       \[\leadsto e^{-x} - \frac{1}{\color{blue}{-\frac{-1}{e^{x} + -2}}} \]

8. Applied egg-rr 4.2%

   \[\leadsto \color{blue}{e^{x} - \left(e^{x} + -2\right)} \]
9. Step-by-step derivation

   1. associate--r+ 4.2%

      \[\leadsto \color{blue}{\left(e^{x} - e^{x}\right) - -2} \]

   2. +-inverses 4.2%

      \[\leadsto \color{blue}{0} - -2 \]

   3. metadata-eval 4.2%

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

10. Simplified 4.2%

    \[\leadsto \color{blue}{2} \]
11. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sinh \left(\frac{x}{2}\right)\\ 4 \cdot \left(t\_0 \cdot t\_0\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sinh (/ x 2.0)))) (* 4.0 (* t_0 t_0))))

C

double code(double x) {
	double t_0 = sinh((x / 2.0));
	return 4.0 * (t_0 * t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = sinh((x / 2.0d0))
    code = 4.0d0 * (t_0 * t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sinh((x / 2.0));
	return 4.0 * (t_0 * t_0);
}

Python

def code(x):
	t_0 = math.sinh((x / 2.0))
	return 4.0 * (t_0 * t_0)

Julia

function code(x)
	t_0 = sinh(Float64(x / 2.0))
	return Float64(4.0 * Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sinh((x / 2.0));
	tmp = 4.0 * (t_0 * t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sinh[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(4.0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]

Reproduce

herbie shell --seed 2024149 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

  :alt
  (! :herbie-platform default (* 4 (* (sinh (/ x 2)) (sinh (/ x 2)))))

  (+ (- (exp x) 2.0) (exp (- x))))