exp2 (problem 3.3.7)

Percentage Accurate: 76.6% → 99.8%

Time: 5.6s

Alternatives: 8

Speedup: 1.9×

• 50.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

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: 76.6% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t_0 \leq 4 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(t_0 + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 4e-10)
     (fma x x (* 0.08333333333333333 (pow x 4.0)))
     (+ (exp x) (+ t_0 -2.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (((exp(x) - 2.0) + t_0) <= 4e-10) {
		tmp = fma(x, x, (0.08333333333333333 * pow(x, 4.0)));
	} else {
		tmp = exp(x) + (t_0 + -2.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + t_0) <= 4e-10)
		tmp = fma(x, x, Float64(0.08333333333333333 * (x ^ 4.0)));
	else
		tmp = Float64(exp(x) + Float64(t_0 + -2.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + t$95$0), $MachinePrecision], 4e-10], N[(x * x + N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Exp[x], $MachinePrecision] + N[(t$95$0 + -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 4.00000000000000015e-10

   1. Initial program 54.7%

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

      1. associate-+l- 54.5%

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

      2. sub-neg 54.5%

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

      3. sub-neg 54.5%

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

      4. +-commutative 54.5%

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

      5. sub0-neg 54.5%

         \[\leadsto e^{x} + \left(-\left(\color{blue}{\left(0 - e^{-x}\right)} + 2\right)\right) \]

      6. associate--r- 54.5%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(0 - \left(e^{-x} - 2\right)\right)}\right) \]

      7. neg-sub0 54.5%

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

      8. remove-double-neg 54.5%

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

      9. sub-neg 54.5%

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

      10. metadata-eval 54.5%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. +-commutative 100.0%

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

      2. unpow2 100.0%

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

      3. fma-def 100.0%

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

   6. Applied egg-rr 100.0%

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

   if 4.00000000000000015e-10 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto e^{x} + \left(-\left(\color{blue}{\left(0 - e^{-x}\right)} + 2\right)\right) \]

      6. associate--r- 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(0 - \left(e^{-x} - 2\right)\right)}\right) \]

      7. neg-sub0 100.0%

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

      8. remove-double-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 4 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t_0 \leq 4 \cdot 10^{-10}:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(t_0 + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 4e-10)
     (pow x 2.0)
     (+ (exp x) (+ t_0 -2.0)))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (((exp(x) - 2.0) + t_0) <= 4e-10) {
		tmp = pow(x, 2.0);
	} else {
		tmp = exp(x) + (t_0 + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if (((exp(x) - 2.0d0) + t_0) <= 4d-10) then
        tmp = x ** 2.0d0
    else
        tmp = exp(x) + (t_0 + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	double tmp;
	if (((Math.exp(x) - 2.0) + t_0) <= 4e-10) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = Math.exp(x) + (t_0 + -2.0);
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if ((math.exp(x) - 2.0) + t_0) <= 4e-10:
		tmp = math.pow(x, 2.0)
	else:
		tmp = math.exp(x) + (t_0 + -2.0)
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + t_0) <= 4e-10)
		tmp = x ^ 2.0;
	else
		tmp = Float64(exp(x) + Float64(t_0 + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = exp(-x);
	tmp = 0.0;
	if (((exp(x) - 2.0) + t_0) <= 4e-10)
		tmp = x ^ 2.0;
	else
		tmp = exp(x) + (t_0 + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + t$95$0), $MachinePrecision], 4e-10], N[Power[x, 2.0], $MachinePrecision], N[(N[Exp[x], $MachinePrecision] + N[(t$95$0 + -2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 4.00000000000000015e-10

   1. Initial program 54.7%

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

      1. associate-+l- 54.5%

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

      2. sub-neg 54.5%

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

      3. sub-neg 54.5%

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

      4. +-commutative 54.5%

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

      5. sub0-neg 54.5%

         \[\leadsto e^{x} + \left(-\left(\color{blue}{\left(0 - e^{-x}\right)} + 2\right)\right) \]

      6. associate--r- 54.5%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(0 - \left(e^{-x} - 2\right)\right)}\right) \]

      7. neg-sub0 54.5%

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

      8. remove-double-neg 54.5%

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

      9. sub-neg 54.5%

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

      10. metadata-eval 54.5%

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

   3. Simplified 54.5%

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

   4. Taylor expanded in x around 0 99.4%

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

   if 4.00000000000000015e-10 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto e^{x} + \left(-\left(\color{blue}{\left(0 - e^{-x}\right)} + 2\right)\right) \]

      6. associate--r- 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(0 - \left(e^{-x} - 2\right)\right)}\right) \]

      7. neg-sub0 100.0%

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

      8. remove-double-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{e^{x} + \left(e^{-x} + -2\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 4 \cdot 10^{-10}:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]

Alternative 3: 87.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x} \cdot 2 - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.1) (pow x 2.0) (- (* (exp x) 2.0) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = pow(x, 2.0);
	} else {
		tmp = (exp(x) * 2.0) - 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.1d0) then
        tmp = x ** 2.0d0
    else
        tmp = (exp(x) * 2.0d0) - 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = (Math.exp(x) * 2.0) - 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.1:
		tmp = math.pow(x, 2.0)
	else:
		tmp = (math.exp(x) * 2.0) - 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.1)
		tmp = x ^ 2.0;
	else
		tmp = Float64(Float64(exp(x) * 2.0) - 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.1)
		tmp = x ^ 2.0;
	else
		tmp = (exp(x) * 2.0) - 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.1], N[Power[x, 2.0], $MachinePrecision], N[(N[(N[Exp[x], $MachinePrecision] * 2.0), $MachinePrecision] - 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000009

   1. Initial program 73.4%

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

      1. associate-+l- 73.3%

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

      2. sub-neg 73.3%

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

      3. sub-neg 73.3%

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

      4. +-commutative 73.3%

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

      5. sub0-neg 73.3%

         \[\leadsto e^{x} + \left(-\left(\color{blue}{\left(0 - e^{-x}\right)} + 2\right)\right) \]

      6. associate--r- 73.3%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(0 - \left(e^{-x} - 2\right)\right)}\right) \]

      7. neg-sub0 73.3%

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

      8. remove-double-neg 73.3%

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

      9. sub-neg 73.3%

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

      10. metadata-eval 73.3%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in x around 0 77.2%

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

   if 3.10000000000000009 < x

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto e^{x} + \left(-\left(\color{blue}{\left(0 - e^{-x}\right)} + 2\right)\right) \]

      6. associate--r- 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(0 - \left(e^{-x} - 2\right)\right)}\right) \]

      7. neg-sub0 100.0%

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

      8. remove-double-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

      1. associate-+r+ 100.0%

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

      2. +-commutative 100.0%

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

      3. associate-+r+ 100.0%

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

      4. metadata-eval 100.0%

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

      5. sub-neg 100.0%

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

      6. associate-+r- 100.0%

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

      7. add-sqr-sqrt 0.0%

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

      8. sqrt-unprod 100.0%

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

      9. sqr-neg 100.0%

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

      10. sqrt-unprod 100.0%

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

      11. add-sqr-sqrt 100.0%

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

      12. count-2 100.0%

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

      13. *-commutative 100.0%

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

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{e^{x} \cdot 2 - 2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x} \cdot 2 - 2\\ \end{array} \]

Alternative 4: 81.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 3.5) (pow x 2.0) (* 0.08333333333333333 (pow x 4.0))))

C

double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = pow(x, 2.0);
	} else {
		tmp = 0.08333333333333333 * pow(x, 4.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.5d0) then
        tmp = x ** 2.0d0
    else
        tmp = 0.08333333333333333d0 * (x ** 4.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.5) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = 0.08333333333333333 * Math.pow(x, 4.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.5:
		tmp = math.pow(x, 2.0)
	else:
		tmp = 0.08333333333333333 * math.pow(x, 4.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.5)
		tmp = x ^ 2.0;
	else
		tmp = Float64(0.08333333333333333 * (x ^ 4.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.5)
		tmp = x ^ 2.0;
	else
		tmp = 0.08333333333333333 * (x ^ 4.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.5], N[Power[x, 2.0], $MachinePrecision], N[(0.08333333333333333 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.5

   1. Initial program 73.4%

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

      1. associate-+l- 73.3%

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

      2. sub-neg 73.3%

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

      3. sub-neg 73.3%

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

      4. +-commutative 73.3%

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

      5. sub0-neg 73.3%

         \[\leadsto e^{x} + \left(-\left(\color{blue}{\left(0 - e^{-x}\right)} + 2\right)\right) \]

      6. associate--r- 73.3%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(0 - \left(e^{-x} - 2\right)\right)}\right) \]

      7. neg-sub0 73.3%

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

      8. remove-double-neg 73.3%

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

      9. sub-neg 73.3%

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

      10. metadata-eval 73.3%

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

   3. Simplified 73.3%

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

   4. Taylor expanded in x around 0 77.2%

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

   if 3.5 < x

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. sub-neg 100.0%

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

      3. sub-neg 100.0%

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

      4. +-commutative 100.0%

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

      5. sub0-neg 100.0%

         \[\leadsto e^{x} + \left(-\left(\color{blue}{\left(0 - e^{-x}\right)} + 2\right)\right) \]

      6. associate--r- 100.0%

         \[\leadsto e^{x} + \left(-\color{blue}{\left(0 - \left(e^{-x} - 2\right)\right)}\right) \]

      7. neg-sub0 100.0%

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

      8. remove-double-neg 100.0%

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

      9. sub-neg 100.0%

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

      10. metadata-eval 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 80.2%

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

   5. Taylor expanded in x around inf 80.2%

      \[\leadsto \color{blue}{0.08333333333333333 \cdot {x}^{4}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.5:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;0.08333333333333333 \cdot {x}^{4}\\ \end{array} \]

Alternative 5: 75.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow x 2.0))

C

double code(double x) {
	return pow(x, 2.0);
}

Fortran

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

Java

public static double code(double x) {
	return Math.pow(x, 2.0);
}

Python

def code(x):
	return math.pow(x, 2.0)

Julia

function code(x)
	return x ^ 2.0
end

MATLAB

function tmp = code(x)
	tmp = x ^ 2.0;
end

Wolfram

code[x_] := N[Power[x, 2.0], $MachinePrecision]

Derivation

1. Initial program 80.3%

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

   1. associate-+l- 80.3%

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

   2. sub-neg 80.3%

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

   3. sub-neg 80.3%

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

   4. +-commutative 80.3%

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

   5. sub0-neg 80.3%

      \[\leadsto e^{x} + \left(-\left(\color{blue}{\left(0 - e^{-x}\right)} + 2\right)\right) \]

   6. associate--r- 80.3%

      \[\leadsto e^{x} + \left(-\color{blue}{\left(0 - \left(e^{-x} - 2\right)\right)}\right) \]

   7. neg-sub0 80.3%

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

   8. remove-double-neg 80.3%

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

   9. sub-neg 80.3%

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

   10. metadata-eval 80.3%

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

3. Simplified 80.3%

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

4. Taylor expanded in x around 0 70.2%

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

5. Final simplification 70.2%

   \[\leadsto {x}^{2} \]

Alternative 6: 29.7% accurate, 41.2× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x (+ x 2.0)))

C

double code(double x) {
	return x * (x + 2.0);
}

Fortran

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

Java

public static double code(double x) {
	return x * (x + 2.0);
}

Python

def code(x):
	return x * (x + 2.0)

Julia

function code(x)
	return Float64(x * Float64(x + 2.0))
end

MATLAB

function tmp = code(x)
	tmp = x * (x + 2.0);
end

Wolfram

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

Derivation

1. Initial program 80.3%

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

   1. associate-+l- 80.3%

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

   2. sub-neg 80.3%

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

   3. sub-neg 80.3%

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

   4. +-commutative 80.3%

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

   5. sub0-neg 80.3%

      \[\leadsto e^{x} + \left(-\left(\color{blue}{\left(0 - e^{-x}\right)} + 2\right)\right) \]

   6. associate--r- 80.3%

      \[\leadsto e^{x} + \left(-\color{blue}{\left(0 - \left(e^{-x} - 2\right)\right)}\right) \]

   7. neg-sub0 80.3%

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

   8. remove-double-neg 80.3%

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

   9. sub-neg 80.3%

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

   10. metadata-eval 80.3%

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

3. Simplified 80.3%

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

   1. associate-+r+ 80.3%

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

   2. +-commutative 80.3%

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

   3. associate-+r+ 80.3%

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

   4. metadata-eval 80.3%

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

   5. sub-neg 80.3%

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

   6. associate-+r- 80.3%

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

   7. add-sqr-sqrt 45.1%

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

   8. sqrt-unprod 80.2%

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

   9. sqr-neg 80.2%

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

   10. sqrt-unprod 35.1%

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

   11. add-sqr-sqrt 49.6%

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

   12. count-2 49.6%

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

   13. *-commutative 49.6%

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

5. Applied egg-rr 49.6%

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

6. Taylor expanded in x around 0 29.8%

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

   1. unpow2 29.8%

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

   2. distribute-rgt-out 29.8%

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

   3. +-commutative 29.8%

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

8. Simplified 29.8%

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

9. Final simplification 29.8%

   \[\leadsto x \cdot \left(x + 2\right) \]

Alternative 7: 75.6% accurate, 41.2× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\frac{1}{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (/ 1.0 x)))

C

double code(double x) {
	return x / (1.0 / x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (1.0d0 / x)
end function

Java

public static double code(double x) {
	return x / (1.0 / x);
}

Python

def code(x):
	return x / (1.0 / x)

Julia

function code(x)
	return Float64(x / Float64(1.0 / x))
end

MATLAB

function tmp = code(x)
	tmp = x / (1.0 / x);
end

Wolfram

code[x_] := N[(x / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

   1. associate-+l- 80.3%

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

   2. sub-neg 80.3%

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

   3. sub-neg 80.3%

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

   4. +-commutative 80.3%

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

   5. sub0-neg 80.3%

      \[\leadsto e^{x} + \left(-\left(\color{blue}{\left(0 - e^{-x}\right)} + 2\right)\right) \]

   6. associate--r- 80.3%

      \[\leadsto e^{x} + \left(-\color{blue}{\left(0 - \left(e^{-x} - 2\right)\right)}\right) \]

   7. neg-sub0 80.3%

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

   8. remove-double-neg 80.3%

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

   9. sub-neg 80.3%

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

   10. metadata-eval 80.3%

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

3. Simplified 80.3%

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

   1. associate-+r+ 80.3%

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

   2. +-commutative 80.3%

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

   3. associate-+r+ 80.3%

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

   4. metadata-eval 80.3%

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

   5. sub-neg 80.3%

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

   6. associate-+r- 80.3%

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

   7. add-sqr-sqrt 45.1%

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

   8. sqrt-unprod 80.2%

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

   9. sqr-neg 80.2%

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

   10. sqrt-unprod 35.1%

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

   11. add-sqr-sqrt 49.6%

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

   12. count-2 49.6%

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

   13. *-commutative 49.6%

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

5. Applied egg-rr 49.6%

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

6. Taylor expanded in x around 0 29.8%

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

   1. unpow2 29.8%

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

   2. distribute-rgt-out 29.8%

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

   3. +-commutative 29.8%

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

8. Simplified 29.8%

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

   1. flip3-+ 16.3%

      \[\leadsto x \cdot \color{blue}{\frac{{x}^{3} + {2}^{3}}{x \cdot x + \left(2 \cdot 2 - x \cdot 2\right)}} \]

   2. associate-*r/ 22.6%

      \[\leadsto \color{blue}{\frac{x \cdot \left({x}^{3} + {2}^{3}\right)}{x \cdot x + \left(2 \cdot 2 - x \cdot 2\right)}} \]

   3. associate-/l* 16.3%

      \[\leadsto \color{blue}{\frac{x}{\frac{x \cdot x + \left(2 \cdot 2 - x \cdot 2\right)}{{x}^{3} + {2}^{3}}}} \]

   4. clear-num 16.3%

      \[\leadsto \frac{x}{\color{blue}{\frac{1}{\frac{{x}^{3} + {2}^{3}}{x \cdot x + \left(2 \cdot 2 - x \cdot 2\right)}}}} \]

   5. flip3-+ 29.8%

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

10. Applied egg-rr 29.8%

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

11. Taylor expanded in x around inf 70.2%

    \[\leadsto \frac{x}{\color{blue}{\frac{1}{x}}} \]

12. Final simplification 70.2%

    \[\leadsto \frac{x}{\frac{1}{x}} \]

Alternative 8: 4.4% accurate, 68.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x 2.0))

C

double code(double x) {
	return x * 2.0;
}

Fortran

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

Java

public static double code(double x) {
	return x * 2.0;
}

Python

def code(x):
	return x * 2.0

Julia

function code(x)
	return Float64(x * 2.0)
end

MATLAB

function tmp = code(x)
	tmp = x * 2.0;
end

Wolfram

code[x_] := N[(x * 2.0), $MachinePrecision]

Derivation

1. Initial program 80.3%

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

   1. associate-+l- 80.3%

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

   2. sub-neg 80.3%

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

   3. sub-neg 80.3%

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

   4. +-commutative 80.3%

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

   5. sub0-neg 80.3%

      \[\leadsto e^{x} + \left(-\left(\color{blue}{\left(0 - e^{-x}\right)} + 2\right)\right) \]

   6. associate--r- 80.3%

      \[\leadsto e^{x} + \left(-\color{blue}{\left(0 - \left(e^{-x} - 2\right)\right)}\right) \]

   7. neg-sub0 80.3%

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

   8. remove-double-neg 80.3%

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

   9. sub-neg 80.3%

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

   10. metadata-eval 80.3%

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

3. Simplified 80.3%

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

   1. associate-+r+ 80.3%

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

   2. +-commutative 80.3%

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

   3. associate-+r+ 80.3%

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

   4. metadata-eval 80.3%

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

   5. sub-neg 80.3%

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

   6. associate-+r- 80.3%

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

   7. add-sqr-sqrt 45.1%

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

   8. sqrt-unprod 80.2%

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

   9. sqr-neg 80.2%

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

   10. sqrt-unprod 35.1%

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

   11. add-sqr-sqrt 49.6%

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

   12. count-2 49.6%

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

   13. *-commutative 49.6%

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

5. Applied egg-rr 49.6%

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

6. Taylor expanded in x around 0 4.5%

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

   1. *-commutative 4.5%

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

8. Simplified 4.5%

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

9. Final simplification 4.5%

   \[\leadsto x \cdot 2 \]

Developer target: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 4 \cdot {\sinh \left(\frac{x}{2}\right)}^{2} \end{array} \]

FPCore

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

C

double code(double x) {
	return 4.0 * pow(sinh((x / 2.0)), 2.0);
}

Fortran

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

Java

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

Python

def code(x):
	return 4.0 * math.pow(math.sinh((x / 2.0)), 2.0)

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(4.0 * N[Power[N[Sinh[N[(x / 2.0), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023336 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64

  :herbie-target
  (* 4.0 (pow (sinh (/ x 2.0)) 2.0))

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