exp2 (problem 3.3.7)

Percentage Accurate: 77.4% → 99.3%

Time: 6.0s

Alternatives: 7

Speedup: 2.0×

• 49.7% 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: 77.4% 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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t_0 \leq 0:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} + t_0\right) - 2\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 0.0) (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) <= 0.0) {
		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) <= 0.0d0) 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) <= 0.0) {
		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) <= 0.0:
		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) <= 0.0)
		tmp = x ^ 2.0;
	else
		tmp = Float64(Float64(exp(x) + 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) <= 0.0)
		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], 0.0], N[Power[x, 2.0], $MachinePrecision], N[(N[(N[Exp[x], $MachinePrecision] + t$95$0), $MachinePrecision] - 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 48.7%

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

      1. +-commutative 48.7%

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

      2. associate-+r- 48.7%

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

      3. metadata-eval 48.7%

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

      4. associate--l- 48.7%

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

      5. associate-+r- 48.7%

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

      6. +-commutative 48.7%

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

      7. associate-+r- 48.7%

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

      8. sub-neg 48.7%

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

      9. associate-+r+ 48.7%

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

      10. associate--l+ 48.7%

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

      11. +-commutative 48.7%

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

      12. associate--l+ 48.7%

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

      13. metadata-eval 48.7%

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

      14. metadata-eval 48.7%

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

   3. Simplified 48.7%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 0.0 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

      1. +-commutative 99.9%

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

      2. associate-+r- 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate--l- 100.0%

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

      5. associate-+r- 99.9%

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

      6. +-commutative 99.9%

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

      7. associate-+r- 99.9%

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

      8. sub-neg 99.9%

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

      9. associate-+r+ 100.0%

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

      10. associate--l+ 100.0%

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

      11. +-commutative 100.0%

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

      12. associate--l+ 100.0%

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

      13. metadata-eval 100.0%

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

      14. 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 inf 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} + e^{-x}\right) - 2\\ \end{array} \]

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(e^{x} - 2\right) + e^{-x}\\ \mathbf{if}\;t_0 \leq 0:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ (- (exp x) 2.0) (exp (- x)))))
   (if (<= t_0 0.0) (pow x 2.0) t_0)))

C

double code(double x) {
	double t_0 = (exp(x) - 2.0) + exp(-x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = pow(x, 2.0);
	} else {
		tmp = t_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) - 2.0d0) + exp(-x)
    if (t_0 <= 0.0d0) then
        tmp = x ** 2.0d0
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (Math.exp(x) - 2.0) + Math.exp(-x);
	double tmp;
	if (t_0 <= 0.0) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.exp(x) - 2.0) + math.exp(-x)
	tmp = 0
	if t_0 <= 0.0:
		tmp = math.pow(x, 2.0)
	else:
		tmp = t_0
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Exp[x], $MachinePrecision] - 2.0), $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 0.0], N[Power[x, 2.0], $MachinePrecision], t$95$0]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 48.7%

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

      1. +-commutative 48.7%

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

      2. associate-+r- 48.7%

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

      3. metadata-eval 48.7%

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

      4. associate--l- 48.7%

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

      5. associate-+r- 48.7%

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

      6. +-commutative 48.7%

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

      7. associate-+r- 48.7%

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

      8. sub-neg 48.7%

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

      9. associate-+r+ 48.7%

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

      10. associate--l+ 48.7%

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

      11. +-commutative 48.7%

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

      12. associate--l+ 48.7%

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

      13. metadata-eval 48.7%

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

      14. metadata-eval 48.7%

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

   3. Simplified 48.7%

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

   4. Taylor expanded in x around 0 100.0%

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

   if 0.0 < (+.f64 (-.f64 (exp.f64 x) 2) (exp.f64 (neg.f64 x)))

   1. Initial program 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 0:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\left(e^{x} - 2\right) + e^{-x}\\ \end{array} \]

Alternative 3: 87.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00017:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00017) (pow x 2.0) (+ (exp x) (+ (exp (- x)) -2.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.00017) {
		tmp = pow(x, 2.0);
	} else {
		tmp = exp(x) + (exp(-x) + -2.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.00017d0) then
        tmp = x ** 2.0d0
    else
        tmp = exp(x) + (exp(-x) + (-2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.00017) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = Math.exp(x) + (Math.exp(-x) + -2.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00017:
		tmp = math.pow(x, 2.0)
	else:
		tmp = math.exp(x) + (math.exp(-x) + -2.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00017)
		tmp = x ^ 2.0;
	else
		tmp = Float64(exp(x) + Float64(exp(Float64(-x)) + -2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.00017)
		tmp = x ^ 2.0;
	else
		tmp = exp(x) + (exp(-x) + -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00017], N[Power[x, 2.0], $MachinePrecision], N[(N[Exp[x], $MachinePrecision] + N[(N[Exp[(-x)], $MachinePrecision] + -2.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.7e-4

   1. Initial program 65.0%

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

      1. +-commutative 65.0%

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

      2. associate-+r- 65.1%

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

      3. metadata-eval 65.1%

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

      4. associate--l- 65.1%

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

      5. associate-+r- 65.0%

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

      6. +-commutative 65.0%

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

      7. associate-+r- 65.0%

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

      8. sub-neg 65.0%

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

      9. associate-+r+ 65.0%

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

      10. associate--l+ 65.0%

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

      11. +-commutative 65.0%

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

      12. associate--l+ 65.0%

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

      13. metadata-eval 65.0%

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

      14. metadata-eval 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in x around 0 81.8%

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

   if 1.7e-4 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate--l- 100.0%

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

      5. associate-+r- 100.0%

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

      6. +-commutative 100.0%

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

      7. associate-+r- 100.0%

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

      8. sub-neg 100.0%

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

      9. associate-+r+ 100.0%

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

      10. associate--l+ 100.0%

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

      11. +-commutative 100.0%

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

      12. associate--l+ 100.0%

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

      13. metadata-eval 100.0%

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

      14. 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 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00017:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]

Alternative 4: 87.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.66:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.66) (pow x 2.0) (expm1 x)))

C

double code(double x) {
	double tmp;
	if (x <= 1.66) {
		tmp = pow(x, 2.0);
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.66) {
		tmp = Math.pow(x, 2.0);
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.66:
		tmp = math.pow(x, 2.0)
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.66)
		tmp = x ^ 2.0;
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.66], N[Power[x, 2.0], $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.65999999999999992

   1. Initial program 65.0%

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

      1. +-commutative 65.0%

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

      2. associate-+r- 65.1%

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

      3. metadata-eval 65.1%

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

      4. associate--l- 65.1%

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

      5. associate-+r- 65.0%

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

      6. +-commutative 65.0%

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

      7. associate-+r- 65.0%

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

      8. sub-neg 65.0%

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

      9. associate-+r+ 65.0%

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

      10. associate--l+ 65.0%

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

      11. +-commutative 65.0%

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

      12. associate--l+ 65.0%

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

      13. metadata-eval 65.0%

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

      14. metadata-eval 65.0%

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

   3. Simplified 65.0%

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

   4. Taylor expanded in x around 0 81.8%

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

   if 1.65999999999999992 < x

   1. Initial program 100.0%

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

      1. +-commutative 100.0%

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

      2. associate-+r- 100.0%

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

      3. metadata-eval 100.0%

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

      4. associate--l- 100.0%

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

      5. associate-+r- 100.0%

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

      6. +-commutative 100.0%

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

      7. associate-+r- 100.0%

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

      8. sub-neg 100.0%

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

      9. associate-+r+ 100.0%

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

      10. associate--l+ 100.0%

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

      11. +-commutative 100.0%

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

      12. associate--l+ 100.0%

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

      13. metadata-eval 100.0%

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

      14. 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 99.1%

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

   5. Taylor expanded in x around inf 99.1%

      \[\leadsto \color{blue}{e^{x} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 99.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]

   7. Simplified 99.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.66:\\ \;\;\;\;{x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 5: 52.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-103}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.2e-103) 0.0 (expm1 x)))

C

double code(double x) {
	double tmp;
	if (x <= 2.2e-103) {
		tmp = 0.0;
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2e-103) {
		tmp = 0.0;
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2e-103:
		tmp = 0.0
	else:
		tmp = math.expm1(x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2e-103)
		tmp = 0.0;
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.2e-103], 0.0, N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.1999999999999999e-103

   1. Initial program 73.4%

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

      1. +-commutative 73.4%

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

      2. associate-+r- 73.4%

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

      3. metadata-eval 73.4%

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

      4. associate--l- 73.4%

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

      5. associate-+r- 73.4%

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

      6. +-commutative 73.4%

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

      7. associate-+r- 73.4%

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

      8. sub-neg 73.4%

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

      9. associate-+r+ 73.4%

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

      10. associate--l+ 73.4%

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

      11. +-commutative 73.4%

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

      12. associate--l+ 73.4%

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

      13. metadata-eval 73.4%

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

      14. metadata-eval 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in x around 0 37.3%

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

   6. Applied egg-rr 37.7%

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

   if 2.1999999999999999e-103 < x

   1. Initial program 75.0%

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

      1. +-commutative 75.0%

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

      2. associate-+r- 75.0%

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

      3. metadata-eval 75.0%

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

      4. associate--l- 75.0%

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

      5. associate-+r- 75.0%

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

      6. +-commutative 75.0%

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

      7. associate-+r- 75.0%

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

      8. sub-neg 75.0%

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

      9. associate-+r+ 75.0%

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

      10. associate--l+ 75.0%

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

      11. +-commutative 75.0%

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

      12. associate--l+ 75.0%

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

      13. metadata-eval 75.0%

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

      14. metadata-eval 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in x around 0 74.3%

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

   5. Taylor expanded in x around inf 74.3%

      \[\leadsto \color{blue}{e^{x} - 1} \]
   6. Step-by-step derivation

      1. expm1-def 75.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]

   7. Simplified 75.1%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-103}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 6: 28.5% accurate, 67.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-103}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.2e-103) 0.0 x))

C

double code(double x) {
	double tmp;
	if (x <= 2.2e-103) {
		tmp = 0.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.2d-103) then
        tmp = 0.0d0
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2e-103) {
		tmp = 0.0;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2e-103:
		tmp = 0.0
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2e-103)
		tmp = 0.0;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2e-103)
		tmp = 0.0;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2e-103], 0.0, x]

Derivation

1. Split input into 2 regimes

2. if x < 2.1999999999999999e-103

   1. Initial program 73.4%

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

      1. +-commutative 73.4%

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

      2. associate-+r- 73.4%

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

      3. metadata-eval 73.4%

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

      4. associate--l- 73.4%

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

      5. associate-+r- 73.4%

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

      6. +-commutative 73.4%

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

      7. associate-+r- 73.4%

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

      8. sub-neg 73.4%

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

      9. associate-+r+ 73.4%

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

      10. associate--l+ 73.4%

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

      11. +-commutative 73.4%

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

      12. associate--l+ 73.4%

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

      13. metadata-eval 73.4%

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

      14. metadata-eval 73.4%

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

   3. Simplified 73.4%

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

   4. Taylor expanded in x around 0 37.3%

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

   6. Applied egg-rr 37.7%

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

   if 2.1999999999999999e-103 < x

   1. Initial program 75.0%

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

      1. +-commutative 75.0%

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

      2. associate-+r- 75.0%

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

      3. metadata-eval 75.0%

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

      4. associate--l- 75.0%

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

      5. associate-+r- 75.0%

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

      6. +-commutative 75.0%

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

      7. associate-+r- 75.0%

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

      8. sub-neg 75.0%

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

      9. associate-+r+ 75.0%

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

      10. associate--l+ 75.0%

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

      11. +-commutative 75.0%

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

      12. associate--l+ 75.0%

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

      13. metadata-eval 75.0%

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

      14. metadata-eval 75.0%

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

   3. Simplified 75.0%

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

   4. Taylor expanded in x around 0 74.3%

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

   5. Taylor expanded in x around 0 5.9%

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

4. Final simplification 26.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2 \cdot 10^{-103}:\\ \;\;\;\;0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 27.2% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 73.9%

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

   1. +-commutative 73.9%

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

   2. associate-+r- 73.9%

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

   3. metadata-eval 73.9%

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

   4. associate--l- 73.9%

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

   5. associate-+r- 73.9%

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

   6. +-commutative 73.9%

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

   7. associate-+r- 73.9%

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

   8. sub-neg 73.9%

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

   9. associate-+r+ 73.9%

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

   10. associate--l+ 73.9%

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

   11. +-commutative 73.9%

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

   12. associate--l+ 73.9%

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

   13. metadata-eval 73.9%

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

   14. metadata-eval 73.9%

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

3. Simplified 73.9%

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

4. Taylor expanded in x around 0 50.0%

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

6. Applied egg-rr 25.5%

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

7. Final simplification 25.5%

   \[\leadsto 0 \]

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 2023301 
(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))))