Hyperbolic sine

Percentage Accurate: 54.4% → 99.7%

Time: 5.2s

Alternatives: 8

Speedup: 18.7×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x} - e^{-x}\\ \mathbf{if}\;t_0 \leq -\infty \lor \neg \left(t_0 \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{t_0}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2 + 0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (exp x) (exp (- x)))))
   (if (or (<= t_0 (- INFINITY)) (not (<= t_0 2e-5)))
     (/ t_0 2.0)
     (/ (+ (* x 2.0) (* 0.3333333333333333 (* x (* x x)))) 2.0))))

C

double code(double x) {
	double t_0 = exp(x) - exp(-x);
	double tmp;
	if ((t_0 <= -((double) INFINITY)) || !(t_0 <= 2e-5)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.exp(x) - Math.exp(-x);
	double tmp;
	if ((t_0 <= -Double.POSITIVE_INFINITY) || !(t_0 <= 2e-5)) {
		tmp = t_0 / 2.0;
	} else {
		tmp = ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0;
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.exp(x) - math.exp(-x)
	tmp = 0
	if (t_0 <= -math.inf) or not (t_0 <= 2e-5):
		tmp = t_0 / 2.0
	else:
		tmp = ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0
	return tmp

Julia

function code(x)
	t_0 = Float64(exp(x) - exp(Float64(-x)))
	tmp = 0.0
	if ((t_0 <= Float64(-Inf)) || !(t_0 <= 2e-5))
		tmp = Float64(t_0 / 2.0);
	else
		tmp = Float64(Float64(Float64(x * 2.0) + Float64(0.3333333333333333 * Float64(x * Float64(x * x)))) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = exp(x) - exp(-x);
	tmp = 0.0;
	if ((t_0 <= -Inf) || ~((t_0 <= 2e-5)))
		tmp = t_0 / 2.0;
	else
		tmp = ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$0, (-Infinity)], N[Not[LessEqual[t$95$0, 2e-5]], $MachinePrecision]], N[(t$95$0 / 2.0), $MachinePrecision], N[(N[(N[(x * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < -inf.0 or 2.00000000000000016e-5 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]

   if -inf.0 < (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2.00000000000000016e-5

   1. Initial program 6.4%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{2} \]
   3. Step-by-step derivation

      1. unpow3 100.0%

         \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{2} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq -\infty \lor \neg \left(e^{x} - e^{-x} \leq 2 \cdot 10^{-5}\right):\\ \;\;\;\;\frac{e^{x} - e^{-x}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2 + 0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{2}\\ \end{array} \]

Alternative 2: 88.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 200000:\\ \;\;\;\;\frac{x \cdot 2 + 0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{6} \cdot 0.027777777777777776}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 200000.0)
   (/ (+ (* x 2.0) (* 0.3333333333333333 (* x (* x x)))) 2.0)
   (sqrt (* (pow x 6.0) 0.027777777777777776))))

C

double code(double x) {
	double tmp;
	if (x <= 200000.0) {
		tmp = ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0;
	} else {
		tmp = sqrt((pow(x, 6.0) * 0.027777777777777776));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 200000.0d0) then
        tmp = ((x * 2.0d0) + (0.3333333333333333d0 * (x * (x * x)))) / 2.0d0
    else
        tmp = sqrt(((x ** 6.0d0) * 0.027777777777777776d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 200000.0) {
		tmp = ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0;
	} else {
		tmp = Math.sqrt((Math.pow(x, 6.0) * 0.027777777777777776));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 200000.0:
		tmp = ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0
	else:
		tmp = math.sqrt((math.pow(x, 6.0) * 0.027777777777777776))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 200000.0)
		tmp = Float64(Float64(Float64(x * 2.0) + Float64(0.3333333333333333 * Float64(x * Float64(x * x)))) / 2.0);
	else
		tmp = sqrt(Float64((x ^ 6.0) * 0.027777777777777776));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 200000.0)
		tmp = ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0;
	else
		tmp = sqrt(((x ^ 6.0) * 0.027777777777777776));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 200000.0], N[(N[(N[(x * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[Sqrt[N[(N[Power[x, 6.0], $MachinePrecision] * 0.027777777777777776), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2e5

   1. Initial program 30.0%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0 89.8%

      \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{2} \]
   3. Step-by-step derivation

      1. unpow3 89.8%

         \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{2} \]

   4. Applied egg-rr 89.8%

      \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{2} \]

   if 2e5 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0 75.6%

      \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{2} \]
   3. Step-by-step derivation

      1. unpow3 75.6%

         \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{2} \]

      2. associate-*r* 75.6%

         \[\leadsto \frac{2 \cdot x + \color{blue}{\left(0.3333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x}}{2} \]

      3. distribute-rgt-out 75.6%

         \[\leadsto \frac{\color{blue}{x \cdot \left(2 + 0.3333333333333333 \cdot \left(x \cdot x\right)\right)}}{2} \]

      4. *-commutative 75.6%

         \[\leadsto \frac{x \cdot \left(2 + \color{blue}{\left(x \cdot x\right) \cdot 0.3333333333333333}\right)}{2} \]

      5. +-commutative 75.6%

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0.3333333333333333 + 2\right)}}{2} \]

      6. associate-*l* 75.6%

         \[\leadsto \frac{x \cdot \left(\color{blue}{x \cdot \left(x \cdot 0.3333333333333333\right)} + 2\right)}{2} \]

      7. fma-def 75.6%

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot 0.3333333333333333, 2\right)}}{2} \]

   4. Simplified 75.6%

      \[\leadsto \frac{\color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot 0.3333333333333333, 2\right)}}{2} \]

   5. Taylor expanded in x around inf 75.6%

      \[\leadsto \frac{x \cdot \color{blue}{\left(0.3333333333333333 \cdot {x}^{2}\right)}}{2} \]
   6. Step-by-step derivation

      1. unpow2 75.6%

         \[\leadsto \frac{x \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot x\right)}\right)}{2} \]

      2. *-commutative 75.6%

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)}}{2} \]

      3. associate-*r* 75.6%

         \[\leadsto \frac{x \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)}}{2} \]

   7. Simplified 75.6%

      \[\leadsto \frac{x \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)}}{2} \]
   8. Step-by-step derivation

      1. add-sqr-sqrt 75.6%

         \[\leadsto \color{blue}{\sqrt{\frac{x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)}{2}} \cdot \sqrt{\frac{x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)}{2}}} \]

      2. sqrt-unprod 88.3%

         \[\leadsto \color{blue}{\sqrt{\frac{x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)}{2} \cdot \frac{x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)}{2}}} \]

      3. div-inv 88.3%

         \[\leadsto \sqrt{\color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) \cdot \frac{1}{2}\right)} \cdot \frac{x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)}{2}} \]

      4. div-inv 88.3%

         \[\leadsto \sqrt{\left(\left(x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) \cdot \frac{1}{2}\right) \cdot \color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) \cdot \frac{1}{2}\right)}} \]

      5. swap-sqr 88.3%

         \[\leadsto \sqrt{\color{blue}{\left(\left(x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)\right) \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2}\right)}} \]

      6. associate-*r* 88.3%

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333\right)\right)} \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2}\right)} \]

      7. associate-*r* 88.3%

         \[\leadsto \sqrt{\left(\color{blue}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.3333333333333333\right)} \cdot \left(x \cdot \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2}\right)} \]

      8. associate-*r* 88.3%

         \[\leadsto \sqrt{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.3333333333333333\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot 0.3333333333333333\right)\right)}\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2}\right)} \]

      9. associate-*r* 88.3%

         \[\leadsto \sqrt{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.3333333333333333\right) \cdot \color{blue}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.3333333333333333\right)}\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2}\right)} \]

      10. swap-sqr 88.3%

          \[\leadsto \sqrt{\color{blue}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)\right)} \cdot \left(\frac{1}{2} \cdot \frac{1}{2}\right)} \]

      11. pow3 88.3%

          \[\leadsto \sqrt{\left(\left(\color{blue}{{x}^{3}} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2}\right)} \]

      12. pow3 88.3%

          \[\leadsto \sqrt{\left(\left({x}^{3} \cdot \color{blue}{{x}^{3}}\right) \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2}\right)} \]

      13. pow-prod-up 88.3%

          \[\leadsto \sqrt{\left(\color{blue}{{x}^{\left(3 + 3\right)}} \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2}\right)} \]

      14. metadata-eval 88.3%

          \[\leadsto \sqrt{\left({x}^{\color{blue}{6}} \cdot \left(0.3333333333333333 \cdot 0.3333333333333333\right)\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2}\right)} \]

      15. metadata-eval 88.3%

          \[\leadsto \sqrt{\left({x}^{6} \cdot \color{blue}{0.1111111111111111}\right) \cdot \left(\frac{1}{2} \cdot \frac{1}{2}\right)} \]

      16. metadata-eval 88.3%

          \[\leadsto \sqrt{\left({x}^{6} \cdot 0.1111111111111111\right) \cdot \left(\color{blue}{0.5} \cdot \frac{1}{2}\right)} \]

      17. metadata-eval 88.3%

          \[\leadsto \sqrt{\left({x}^{6} \cdot 0.1111111111111111\right) \cdot \left(0.5 \cdot \color{blue}{0.5}\right)} \]

      18. metadata-eval 88.3%

          \[\leadsto \sqrt{\left({x}^{6} \cdot 0.1111111111111111\right) \cdot \color{blue}{0.25}} \]

   9. Applied egg-rr 88.3%

      \[\leadsto \color{blue}{\sqrt{\left({x}^{6} \cdot 0.1111111111111111\right) \cdot 0.25}} \]
   10. Step-by-step derivation

       1. associate-*l* 88.3%

          \[\leadsto \sqrt{\color{blue}{{x}^{6} \cdot \left(0.1111111111111111 \cdot 0.25\right)}} \]

       2. metadata-eval 88.3%

          \[\leadsto \sqrt{{x}^{6} \cdot \color{blue}{0.027777777777777776}} \]

   11. Simplified 88.3%

       \[\leadsto \color{blue}{\sqrt{{x}^{6} \cdot 0.027777777777777776}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 200000:\\ \;\;\;\;\frac{x \cdot 2 + 0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{{x}^{6} \cdot 0.027777777777777776}\\ \end{array} \]

Alternative 3: 84.5% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot 2 + 0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (+ (* x 2.0) (* 0.3333333333333333 (* x (* x x)))) 2.0))

C

double code(double x) {
	return ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * 2.0d0) + (0.3333333333333333d0 * (x * (x * x)))) / 2.0d0
end function

Java

public static double code(double x) {
	return ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0;
}

Python

def code(x):
	return ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0

Julia

function code(x)
	return Float64(Float64(Float64(x * 2.0) + Float64(0.3333333333333333 * Float64(x * Float64(x * x)))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = ((x * 2.0) + (0.3333333333333333 * (x * (x * x)))) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[(x * 2.0), $MachinePrecision] + N[(0.3333333333333333 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 48.1%

   \[\frac{e^{x} - e^{-x}}{2} \]

2. Taylor expanded in x around 0 86.2%

   \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{2} \]
3. Step-by-step derivation

   1. unpow3 86.2%

      \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{2} \]

4. Applied egg-rr 86.2%

   \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{2} \]

5. Final simplification 86.2%

   \[\leadsto \frac{x \cdot 2 + 0.3333333333333333 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{2} \]

Alternative 4: 84.1% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.4 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.4) (not (<= x 2.5)))
   (* x (* x (* x 0.16666666666666666)))
   (/ (* x 2.0) 2.0)))

C

double code(double x) {
	double tmp;
	if ((x <= -2.4) || !(x <= 2.5)) {
		tmp = x * (x * (x * 0.16666666666666666));
	} else {
		tmp = (x * 2.0) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.4d0)) .or. (.not. (x <= 2.5d0))) then
        tmp = x * (x * (x * 0.16666666666666666d0))
    else
        tmp = (x * 2.0d0) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -2.4) || !(x <= 2.5)) {
		tmp = x * (x * (x * 0.16666666666666666));
	} else {
		tmp = (x * 2.0) / 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -2.4) or not (x <= 2.5):
		tmp = x * (x * (x * 0.16666666666666666))
	else:
		tmp = (x * 2.0) / 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -2.4) || !(x <= 2.5))
		tmp = Float64(x * Float64(x * Float64(x * 0.16666666666666666)));
	else
		tmp = Float64(Float64(x * 2.0) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -2.4) || ~((x <= 2.5)))
		tmp = x * (x * (x * 0.16666666666666666));
	else
		tmp = (x * 2.0) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -2.4], N[Not[LessEqual[x, 2.5]], $MachinePrecision]], N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.39999999999999991 or 2.5 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0 69.2%

      \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{2} \]
   3. Step-by-step derivation

      1. unpow3 69.2%

         \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{2} \]

      2. associate-*r* 69.2%

         \[\leadsto \frac{2 \cdot x + \color{blue}{\left(0.3333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x}}{2} \]

      3. distribute-rgt-out 69.2%

         \[\leadsto \frac{\color{blue}{x \cdot \left(2 + 0.3333333333333333 \cdot \left(x \cdot x\right)\right)}}{2} \]

      4. *-commutative 69.2%

         \[\leadsto \frac{x \cdot \left(2 + \color{blue}{\left(x \cdot x\right) \cdot 0.3333333333333333}\right)}{2} \]

      5. +-commutative 69.2%

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0.3333333333333333 + 2\right)}}{2} \]

      6. associate-*l* 69.2%

         \[\leadsto \frac{x \cdot \left(\color{blue}{x \cdot \left(x \cdot 0.3333333333333333\right)} + 2\right)}{2} \]

      7. fma-def 69.2%

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot 0.3333333333333333, 2\right)}}{2} \]

   4. Simplified 69.2%

      \[\leadsto \frac{\color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot 0.3333333333333333, 2\right)}}{2} \]

   5. Taylor expanded in x around inf 69.2%

      \[\leadsto \frac{x \cdot \color{blue}{\left(0.3333333333333333 \cdot {x}^{2}\right)}}{2} \]
   6. Step-by-step derivation

      1. unpow2 69.2%

         \[\leadsto \frac{x \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(x \cdot x\right)}\right)}{2} \]

      2. *-commutative 69.2%

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0.3333333333333333\right)}}{2} \]

      3. associate-*r* 69.2%

         \[\leadsto \frac{x \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)}}{2} \]

   7. Simplified 69.2%

      \[\leadsto \frac{x \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.3333333333333333\right)\right)}}{2} \]
   8. Step-by-step derivation

      1. associate-/l* 69.2%

         \[\leadsto \color{blue}{\frac{x}{\frac{2}{x \cdot \left(x \cdot 0.3333333333333333\right)}}} \]

      2. div-inv 69.2%

         \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{2}{x \cdot \left(x \cdot 0.3333333333333333\right)}}} \]

      3. associate-*r* 69.2%

         \[\leadsto x \cdot \frac{1}{\frac{2}{\color{blue}{\left(x \cdot x\right) \cdot 0.3333333333333333}}} \]

      4. *-commutative 69.2%

         \[\leadsto x \cdot \frac{1}{\frac{2}{\color{blue}{0.3333333333333333 \cdot \left(x \cdot x\right)}}} \]

   9. Applied egg-rr 69.2%

      \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{2}{0.3333333333333333 \cdot \left(x \cdot x\right)}}} \]

   10. Taylor expanded in x around 0 69.2%

       \[\leadsto x \cdot \color{blue}{\left(0.16666666666666666 \cdot {x}^{2}\right)} \]
   11. Step-by-step derivation

       1. unpow2 69.2%

          \[\leadsto x \cdot \left(0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

       2. *-commutative 69.2%

          \[\leadsto x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0.16666666666666666\right)} \]

       3. associate-*l* 69.2%

          \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.16666666666666666\right)\right)} \]

   12. Simplified 69.2%

       \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.16666666666666666\right)\right)} \]

   if -2.39999999999999991 < x < 2.5

   1. Initial program 7.1%

      \[\frac{e^{x} - e^{-x}}{2} \]

   2. Taylor expanded in x around 0 99.3%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.4 \lor \neg \left(x \leq 2.5\right):\\ \;\;\;\;x \cdot \left(x \cdot \left(x \cdot 0.16666666666666666\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{2}\\ \end{array} \]

Alternative 5: 84.4% accurate, 18.7× speedup

Math

\[\begin{array}{l} \\ \frac{x \cdot \left(2 + x \cdot \left(x \cdot 0.3333333333333333\right)\right)}{2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* x (+ 2.0 (* x (* x 0.3333333333333333)))) 2.0))

C

double code(double x) {
	return (x * (2.0 + (x * (x * 0.3333333333333333)))) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (2.0d0 + (x * (x * 0.3333333333333333d0)))) / 2.0d0
end function

Java

public static double code(double x) {
	return (x * (2.0 + (x * (x * 0.3333333333333333)))) / 2.0;
}

Python

def code(x):
	return (x * (2.0 + (x * (x * 0.3333333333333333)))) / 2.0

Julia

function code(x)
	return Float64(Float64(x * Float64(2.0 + Float64(x * Float64(x * 0.3333333333333333)))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (x * (2.0 + (x * (x * 0.3333333333333333)))) / 2.0;
end

Wolfram

code[x_] := N[(N[(x * N[(2.0 + N[(x * N[(x * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 48.1%

   \[\frac{e^{x} - e^{-x}}{2} \]

2. Taylor expanded in x around 0 86.2%

   \[\leadsto \frac{\color{blue}{2 \cdot x + 0.3333333333333333 \cdot {x}^{3}}}{2} \]
3. Step-by-step derivation

   1. unpow3 86.2%

      \[\leadsto \frac{2 \cdot x + 0.3333333333333333 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{2} \]

   2. associate-*r* 86.2%

      \[\leadsto \frac{2 \cdot x + \color{blue}{\left(0.3333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x}}{2} \]

   3. distribute-rgt-out 86.1%

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + 0.3333333333333333 \cdot \left(x \cdot x\right)\right)}}{2} \]

   4. *-commutative 86.1%

      \[\leadsto \frac{x \cdot \left(2 + \color{blue}{\left(x \cdot x\right) \cdot 0.3333333333333333}\right)}{2} \]

   5. +-commutative 86.1%

      \[\leadsto \frac{x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot 0.3333333333333333 + 2\right)}}{2} \]

   6. associate-*l* 86.1%

      \[\leadsto \frac{x \cdot \left(\color{blue}{x \cdot \left(x \cdot 0.3333333333333333\right)} + 2\right)}{2} \]

   7. fma-def 86.1%

      \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot 0.3333333333333333, 2\right)}}{2} \]

4. Simplified 86.1%

   \[\leadsto \frac{\color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot 0.3333333333333333, 2\right)}}{2} \]
5. Step-by-step derivation

   1. fma-udef 86.1%

      \[\leadsto \frac{x \cdot \color{blue}{\left(x \cdot \left(x \cdot 0.3333333333333333\right) + 2\right)}}{2} \]

   2. *-commutative 86.1%

      \[\leadsto \frac{x \cdot \left(x \cdot \color{blue}{\left(0.3333333333333333 \cdot x\right)} + 2\right)}{2} \]

6. Applied egg-rr 86.1%

   \[\leadsto \frac{x \cdot \color{blue}{\left(x \cdot \left(0.3333333333333333 \cdot x\right) + 2\right)}}{2} \]

7. Final simplification 86.1%

   \[\leadsto \frac{x \cdot \left(2 + x \cdot \left(x \cdot 0.3333333333333333\right)\right)}{2} \]

Alternative 6: 52.2% accurate, 41.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.1%

   \[\frac{e^{x} - e^{-x}}{2} \]

2. Taylor expanded in x around 0 58.0%

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

3. Final simplification 58.0%

   \[\leadsto \frac{x \cdot 2}{2} \]

Alternative 7: 2.9% accurate, 206.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -1.0)

C

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

Fortran

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

Java

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

Python

def code(x):
	return -1.0

Julia

function code(x)
	return -1.0
end

MATLAB

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

Wolfram

code[x_] := -1.0

Derivation

1. Initial program 48.1%

   \[\frac{e^{x} - e^{-x}}{2} \]

3. Applied egg-rr 2.9%

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

4. Final simplification 2.9%

   \[\leadsto -1 \]

Alternative 8: 3.5% 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 48.1%

   \[\frac{e^{x} - e^{-x}}{2} \]

3. Applied egg-rr 3.7%

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

4. Final simplification 3.7%

   \[\leadsto 0 \]

Reproduce

herbie shell --seed 2023217 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))