exp2 (problem 3.3.7)

Percentage Accurate: 76.4% → 99.8%

Time: 5.4s

Alternatives: 7

Speedup: 18.7×

• 48.8% 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: 76.4% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(e^{x} - 2\right) + t_0 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(t_0 + -2\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (+ (- (exp x) 2.0) t_0) 5e-12)
     (fma x x (* 0.08333333333333333 (* (* x x) (* x x))))
     (+ (exp x) (+ t_0 -2.0)))))

C

x = abs(x);
double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if (((exp(x) - 2.0) + t_0) <= 5e-12) {
		tmp = fma(x, x, (0.08333333333333333 * ((x * x) * (x * x))));
	} else {
		tmp = exp(x) + (t_0 + -2.0);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(Float64(exp(x) - 2.0) + t_0) <= 5e-12)
		tmp = fma(x, x, Float64(0.08333333333333333 * Float64(Float64(x * x) * Float64(x * x))));
	else
		tmp = Float64(exp(x) + Float64(t_0 + -2.0));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
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], 5e-12], N[(x * x + N[(0.08333333333333333 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $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))) < 4.9999999999999997e-12

   1. Initial program 53.6%

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

      1. associate-+l- 53.6%

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

      2. sub-neg 53.6%

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

      3. sub-neg 53.6%

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

      4. +-commutative 53.6%

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

      5. distribute-neg-in 53.6%

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

      6. remove-double-neg 53.6%

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

      7. metadata-eval 53.6%

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

   3. Simplified 53.6%

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

   4. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

      2. fma-def 100.0%

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

   6. Simplified 100.0%

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

      1. sqr-pow 100.0%

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

      2. metadata-eval 100.0%

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

      3. pow2 100.0%

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

      4. metadata-eval 100.0%

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

      5. pow2 100.0%

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

   8. Applied egg-rr 100.0%

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

   if 4.9999999999999997e-12 < (+.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. associate-+l- 99.9%

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

      2. sub-neg 99.9%

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

      3. sub-neg 99.9%

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

      4. +-commutative 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. remove-double-neg 99.9%

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

      7. metadata-eval 99.9%

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

   3. Simplified 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(e^{x} - 2\right) + e^{-x} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{x} + \left(e^{-x} + -2\right)\\ \end{array} \]

Alternative 2: 99.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.6)
   (fma x x (* 0.08333333333333333 (* (* x x) (* x x))))
   (expm1 x)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = fma(x, x, (0.08333333333333333 * ((x * x) * (x * x))));
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.6)
		tmp = fma(x, x, Float64(0.08333333333333333 * Float64(Float64(x * x) * Float64(x * x))));
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.6], N[(x * x + N[(0.08333333333333333 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 70.7%

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

      1. associate-+l- 70.7%

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

      2. sub-neg 70.7%

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

      3. sub-neg 70.7%

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

      4. +-commutative 70.7%

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

      5. distribute-neg-in 70.7%

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

      6. remove-double-neg 70.7%

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

      7. metadata-eval 70.7%

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

   3. Simplified 70.7%

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

   4. Taylor expanded in x around 0 89.7%

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

      1. unpow2 89.7%

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

      2. fma-def 89.7%

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

   6. Simplified 89.7%

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

      1. sqr-pow 89.7%

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

      2. metadata-eval 89.7%

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

      3. pow2 89.7%

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

      4. metadata-eval 89.7%

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

      5. pow2 89.7%

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

   8. Applied egg-rr 89.7%

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

   if 2.60000000000000009 < 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. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. 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 98.8%

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

   5. Taylor expanded in x around inf 98.8%

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

      1. expm1-def 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\mathsf{fma}\left(x, x, 0.08333333333333333 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 3: 99.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.08333333333333333\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.6) (* (* x x) (+ (* x (* x 0.08333333333333333)) 1.0)) (expm1 x)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = (x * x) * ((x * (x * 0.08333333333333333)) + 1.0);
	} else {
		tmp = expm1(x);
	}
	return tmp;
}

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = (x * x) * ((x * (x * 0.08333333333333333)) + 1.0);
	} else {
		tmp = Math.expm1(x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.6:
		tmp = (x * x) * ((x * (x * 0.08333333333333333)) + 1.0)
	else:
		tmp = math.expm1(x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.6)
		tmp = Float64(Float64(x * x) * Float64(Float64(x * Float64(x * 0.08333333333333333)) + 1.0));
	else
		tmp = expm1(x);
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.6], N[(N[(x * x), $MachinePrecision] * N[(N[(x * N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(Exp[x] - 1), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 70.7%

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

      1. associate-+l- 70.7%

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

      2. sub-neg 70.7%

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

      3. sub-neg 70.7%

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

      4. +-commutative 70.7%

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

      5. distribute-neg-in 70.7%

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

      6. remove-double-neg 70.7%

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

      7. metadata-eval 70.7%

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

   3. Simplified 70.7%

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

   4. Taylor expanded in x around 0 89.7%

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

      1. unpow2 89.7%

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

   6. Simplified 89.7%

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

      1. sqr-pow 89.7%

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

      2. metadata-eval 89.7%

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

      3. pow2 89.7%

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

      4. metadata-eval 89.7%

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

      5. pow2 89.7%

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

   8. Applied egg-rr 89.7%

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

      1. associate-*r* 89.7%

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

      2. distribute-rgt1-in 89.7%

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

      3. *-commutative 89.7%

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

   10. Applied egg-rr 89.7%

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

   11. Taylor expanded in x around 0 89.7%

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

       1. unpow2 89.7%

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

       2. *-commutative 89.7%

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

       3. associate-*r* 89.7%

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

       4. *-commutative 89.7%

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

   13. Simplified 89.7%

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

   if 2.60000000000000009 < 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. distribute-neg-in 100.0%

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

      6. remove-double-neg 100.0%

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

      7. 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 98.8%

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

   5. Taylor expanded in x around inf 98.8%

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

      1. expm1-def 98.8%

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

   7. Simplified 98.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.08333333333333333\right) + 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{expm1}\left(x\right)\\ \end{array} \]

Alternative 4: 87.9% accurate, 18.7× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (* (* x x) (+ (* x (* x 0.08333333333333333)) 1.0)))

C

x = abs(x);
double code(double x) {
	return (x * x) * ((x * (x * 0.08333333333333333)) + 1.0);
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * x) * ((x * (x * 0.08333333333333333d0)) + 1.0d0)
end function

Java

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

Python

x = abs(x)
def code(x):
	return (x * x) * ((x * (x * 0.08333333333333333)) + 1.0)

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(N[(x * x), $MachinePrecision] * N[(N[(x * N[(x * 0.08333333333333333), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 77.6%

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

   1. associate-+l- 77.7%

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

   2. sub-neg 77.7%

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

   3. sub-neg 77.7%

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

   4. +-commutative 77.7%

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

   5. distribute-neg-in 77.7%

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

   6. remove-double-neg 77.7%

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

   7. metadata-eval 77.7%

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

3. Simplified 77.7%

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

4. Taylor expanded in x around 0 86.0%

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

   1. unpow2 86.0%

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

6. Simplified 86.0%

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

   1. sqr-pow 86.0%

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

   2. metadata-eval 86.0%

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

   3. pow2 86.0%

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

   4. metadata-eval 86.0%

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

   5. pow2 86.0%

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

8. Applied egg-rr 86.0%

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

   1. associate-*r* 86.0%

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

   2. distribute-rgt1-in 86.0%

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

   3. *-commutative 86.0%

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

10. Applied egg-rr 86.0%

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

11. Taylor expanded in x around 0 86.0%

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

    1. unpow2 86.0%

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

    2. *-commutative 86.0%

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

    3. associate-*r* 86.0%

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

    4. *-commutative 86.0%

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

13. Simplified 86.0%

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

14. Final simplification 86.0%

    \[\leadsto \left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot 0.08333333333333333\right) + 1\right) \]

Alternative 5: 29.8% accurate, 67.3× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 2.2e-103) 0.0 x))

C

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

Fortran

NOTE: x should be positive before calling this function
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

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

Python

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

Julia

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

MATLAB

x = abs(x)
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

NOTE: x should be positive before calling this function
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 79.3%

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

      1. associate-+l- 79.3%

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

      2. sub-neg 79.3%

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

      3. sub-neg 79.3%

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

      4. +-commutative 79.3%

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

      5. distribute-neg-in 79.3%

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

      6. remove-double-neg 79.3%

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

      7. metadata-eval 79.3%

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

   3. Simplified 79.3%

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

   5. Applied egg-rr 38.0%

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

   if 2.1999999999999999e-103 < x

   1. Initial program 74.2%

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

      1. associate-+l- 74.2%

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

      2. sub-neg 74.2%

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

      3. sub-neg 74.2%

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

      4. +-commutative 74.2%

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

      5. distribute-neg-in 74.2%

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

      6. remove-double-neg 74.2%

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

      7. metadata-eval 74.2%

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

   3. Simplified 74.2%

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

   4. Taylor expanded in x around 0 73.0%

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

   5. Taylor expanded in x around 0 6.2%

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

4. Final simplification 27.5%

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

Alternative 6: 76.1% accurate, 68.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ x \cdot x \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (* x x))

C

x = abs(x);
double code(double x) {
	return x * x;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = x * x
end function

Java

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

Python

x = abs(x)
def code(x):
	return x * x

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := N[(x * x), $MachinePrecision]

Derivation

1. Initial program 77.6%

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

   1. associate-+l- 77.7%

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

   2. sub-neg 77.7%

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

   3. sub-neg 77.7%

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

   4. +-commutative 77.7%

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

   5. distribute-neg-in 77.7%

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

   6. remove-double-neg 77.7%

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

   7. metadata-eval 77.7%

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

3. Simplified 77.7%

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

4. Taylor expanded in x around 0 75.8%

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

   1. unpow2 75.8%

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

6. Simplified 75.8%

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

7. Final simplification 75.8%

   \[\leadsto x \cdot x \]

Alternative 7: 27.3% accurate, 206.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 0 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 0.0)

C

x = abs(x);
double code(double x) {
	return 0.0;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 0.0;
}

Python

x = abs(x)
def code(x):
	return 0.0

Julia

x = abs(x)
function code(x)
	return 0.0
end

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 0.0

Derivation

1. Initial program 77.6%

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

   1. associate-+l- 77.7%

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

   2. sub-neg 77.7%

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

   3. sub-neg 77.7%

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

   4. +-commutative 77.7%

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

   5. distribute-neg-in 77.7%

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

   6. remove-double-neg 77.7%

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

   7. metadata-eval 77.7%

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

3. Simplified 77.7%

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

5. Applied egg-rr 26.3%

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

6. Final simplification 26.3%

   \[\leadsto 0 \]

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 2023200 
(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))))