Hyperbolic tangent

Percentage Accurate: 9.1% → 100.0%

Time: 13.6s

Alternatives: 9

Speedup: 422.0×

• could not determine a ground truth

  1. x = -5.328241899185679e+279

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \frac{e^{x} - t\_0}{e^{x} + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))) (/ (- (exp x) t_0) (+ (exp x) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	return (exp(x) - t_0) / (exp(x) + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = exp(-x)
    code = (exp(x) - t_0) / (exp(x) + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	return (Math.exp(x) - t_0) / (Math.exp(x) + t_0);
}

Python

def code(x):
	t_0 = math.exp(-x)
	return (math.exp(x) - t_0) / (math.exp(x) + t_0)

Julia

function code(x)
	t_0 = exp(Float64(-x))
	return Float64(Float64(exp(x) - t_0) / Float64(exp(x) + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = exp(-x);
	tmp = (exp(x) - t_0) / (exp(x) + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, N[(N[(N[Exp[x], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 9.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \frac{e^{x} - t\_0}{e^{x} + t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))) (/ (- (exp x) t_0) (+ (exp x) t_0))))

C

double code(double x) {
	double t_0 = exp(-x);
	return (exp(x) - t_0) / (exp(x) + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = exp(-x)
    code = (exp(x) - t_0) / (exp(x) + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.exp(-x);
	return (Math.exp(x) - t_0) / (Math.exp(x) + t_0);
}

Python

def code(x):
	t_0 = math.exp(-x)
	return (math.exp(x) - t_0) / (math.exp(x) + t_0)

Julia

function code(x)
	t_0 = exp(Float64(-x))
	return Float64(Float64(exp(x) - t_0) / Float64(exp(x) + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = exp(-x);
	tmp = (exp(x) - t_0) / (exp(x) + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, N[(N[(N[Exp[x], $MachinePrecision] - t$95$0), $MachinePrecision] / N[(N[Exp[x], $MachinePrecision] + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 100.0% accurate, 4.2× speedup

Math

\[\begin{array}{l} \\ \tanh x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (tanh x))

C

double code(double x) {
	return tanh(x);
}

Fortran

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

Java

public static double code(double x) {
	return Math.tanh(x);
}

Python

def code(x):
	return math.tanh(x)

Julia

function code(x)
	return tanh(x)
end

MATLAB

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

Wolfram

code[x_] := N[Tanh[x], $MachinePrecision]

Derivation

1. Initial program 10.3%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. tanh-undef N/A

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

   2. lower-tanh.f64 100.0

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

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\tanh x} \]
5. Add Preprocessing

Alternative 2: 97.6% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma
   (fma
    (* x x)
    (fma x (* x 0.0001984126984126984) 0.008333333333333333)
    0.16666666666666666)
   (* x (* x x))
   x)
  (fma
   (* x x)
   (fma (* x x) (fma (* x x) 0.001388888888888889 0.041666666666666664) 0.5)
   1.0)))

C

double code(double x) {
	return fma(fma((x * x), fma(x, (x * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666), (x * (x * x)), x) / fma((x * x), fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0);
}

Julia

function code(x)
	return Float64(fma(fma(Float64(x * x), fma(x, Float64(x * 0.0001984126984126984), 0.008333333333333333), 0.16666666666666666), Float64(x * Float64(x * x)), x) / fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0))
end

Wolfram

code[x_] := N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 0.0001984126984126984), $MachinePrecision] + 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.3%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift--.f64 N/A

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

   5. cosh-undef N/A

      \[\leadsto \frac{e^{x} - e^{\mathsf{neg}\left(x\right)}}{\color{blue}{2 \cdot \cosh x}} \]

   6. associate-/r* N/A

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

   7. lift--.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. sinh-def N/A

       \[\leadsto \frac{\color{blue}{\sinh x}}{\cosh x} \]

   12. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\sinh x}{\cosh x}} \]

   13. lower-sinh.f64 N/A

       \[\leadsto \frac{\color{blue}{\sinh x}}{\cosh x} \]

   14. lower-cosh.f64 98.0

       \[\leadsto \frac{\sinh x}{\color{blue}{\cosh x}} \]

4. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\sinh x}{\cosh x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\sinh x}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\sinh x}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{\sinh x}{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)}} \]

   3. unpow2 N/A

      \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, 1\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right)}, 1\right)} \]

   7. unpow2 N/A

      \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right)} \]

   9. +-commutative N/A

      \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{1}{2}\right), 1\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right), 1\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right)} \]

   12. unpow2 N/A

       \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   13. lower-*.f64 97.9

       \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]

7. Applied rewrites 97.9%

   \[\leadsto \frac{\sinh x}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) \cdot x}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   2. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   3. distribute-lft1-in N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) \cdot x + x}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   4. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right)} \cdot x + x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) \cdot \left({x}^{2} \cdot x\right)} + x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   6. unpow2 N/A

      \[\leadsto \frac{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) + x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   7. unpow3 N/A

      \[\leadsto \frac{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) \cdot \color{blue}{{x}^{3}} + x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), {x}^{3}, x\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

10. Applied rewrites 98.1%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
11. Add Preprocessing

Alternative 3: 97.5% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fma (fma x (* x 0.008333333333333333) 0.16666666666666666) (* x (* x x)) x)
  (fma
   (* x x)
   (fma (* x x) (fma (* x x) 0.001388888888888889 0.041666666666666664) 0.5)
   1.0)))

C

double code(double x) {
	return fma(fma(x, (x * 0.008333333333333333), 0.16666666666666666), (x * (x * x)), x) / fma((x * x), fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0);
}

Julia

function code(x)
	return Float64(fma(fma(x, Float64(x * 0.008333333333333333), 0.16666666666666666), Float64(x * Float64(x * x)), x) / fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0))
end

Wolfram

code[x_] := N[(N[(N[(x * N[(x * 0.008333333333333333), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.3%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-neg.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift--.f64 N/A

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

   5. cosh-undef N/A

      \[\leadsto \frac{e^{x} - e^{\mathsf{neg}\left(x\right)}}{\color{blue}{2 \cdot \cosh x}} \]

   6. associate-/r* N/A

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

   7. lift--.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-neg.f64 N/A

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

   11. sinh-def N/A

       \[\leadsto \frac{\color{blue}{\sinh x}}{\cosh x} \]

   12. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\sinh x}{\cosh x}} \]

   13. lower-sinh.f64 N/A

       \[\leadsto \frac{\color{blue}{\sinh x}}{\cosh x} \]

   14. lower-cosh.f64 98.0

       \[\leadsto \frac{\sinh x}{\color{blue}{\cosh x}} \]

4. Applied rewrites 98.0%

   \[\leadsto \color{blue}{\frac{\sinh x}{\cosh x}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{\sinh x}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{\sinh x}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{\sinh x}{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)}} \]

   3. unpow2 N/A

      \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)} \]

   5. +-commutative N/A

      \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, 1\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right)}, 1\right)} \]

   7. unpow2 N/A

      \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right)} \]

   8. lower-*.f64 N/A

      \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right)} \]

   9. +-commutative N/A

      \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{1}{2}\right), 1\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right), 1\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right)} \]

   12. unpow2 N/A

       \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   13. lower-*.f64 97.9

       \[\leadsto \frac{\sinh x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]

7. Applied rewrites 97.9%

   \[\leadsto \frac{\sinh x}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]
9. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) + 1\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x + 1 \cdot x}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot {x}^{2}\right)} \cdot x + 1 \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   4. associate-*l* N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot x\right)} + 1 \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   5. unpow2 N/A

      \[\leadsto \frac{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) + 1 \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   6. unpow3 N/A

      \[\leadsto \frac{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{3}} + 1 \cdot x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   7. *-lft-identity N/A

      \[\leadsto \frac{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot {x}^{3} + \color{blue}{x}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}, {x}^{3}, x\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {x}^{2} + \frac{1}{6}}, {x}^{3}, x\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   10. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\frac{1}{120} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{6}, {x}^{3}, x\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   11. associate-*r* N/A

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{120} \cdot x\right) \cdot x} + \frac{1}{6}, {x}^{3}, x\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   12. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{1}{120} \cdot x\right)} + \frac{1}{6}, {x}^{3}, x\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, \frac{1}{120} \cdot x, \frac{1}{6}\right)}, {x}^{3}, x\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   14. *-commutative N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{120}}, \frac{1}{6}\right), {x}^{3}, x\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{120}}, \frac{1}{6}\right), {x}^{3}, x\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   16. cube-mult N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{1}{120}, \frac{1}{6}\right), \color{blue}{x \cdot \left(x \cdot x\right)}, x\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   17. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{1}{120}, \frac{1}{6}\right), x \cdot \color{blue}{{x}^{2}}, x\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   18. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{1}{120}, \frac{1}{6}\right), \color{blue}{x \cdot {x}^{2}}, x\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   19. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{1}{120}, \frac{1}{6}\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right)} \]

   20. lower-*.f64 98.0

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]

10. Applied rewrites 98.0%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.008333333333333333, 0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)}}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \]
11. Add Preprocessing

Alternative 4: 97.2% accurate, 10.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (fma
   x
   (* x (fma x (* x -0.05396825396825397) 0.13333333333333333))
   -0.3333333333333333)
  (* x (* x x))
  x))

C

double code(double x) {
	return fma(fma(x, (x * fma(x, (x * -0.05396825396825397), 0.13333333333333333)), -0.3333333333333333), (x * (x * x)), x);
}

Julia

function code(x)
	return fma(fma(x, Float64(x * fma(x, Float64(x * -0.05396825396825397), 0.13333333333333333)), -0.3333333333333333), Float64(x * Float64(x * x)), x)
end

Wolfram

code[x_] := N[(N[(x * N[(x * N[(x * N[(x * -0.05396825396825397), $MachinePrecision] + 0.13333333333333333), $MachinePrecision]), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 10.3%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right) \cdot x + 1 \cdot x} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} + 1 \cdot x \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)} + 1 \cdot x \]

   5. *-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) \cdot \left(x \cdot {x}^{2}\right)} + 1 \cdot x \]

   6. *-lft-identity N/A

      \[\leadsto \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right) \cdot \left(x \cdot {x}^{2}\right) + \color{blue}{x} \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}, x \cdot {x}^{2}, x\right)} \]

5. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot -0.05396825396825397, 0.13333333333333333\right), -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]
6. Add Preprocessing

Alternative 5: 97.4% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right), x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (* (* x x) (fma (* x x) 0.13333333333333333 -0.3333333333333333)) x x))

C

double code(double x) {
	return fma(((x * x) * fma((x * x), 0.13333333333333333, -0.3333333333333333)), x, x);
}

Julia

function code(x)
	return fma(Float64(Float64(x * x) * fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333)), x, x)
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 10.3%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) \cdot x + 1 \cdot x} \]

   3. *-lft-identity N/A

      \[\leadsto \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) \cdot x + \color{blue}{x} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} + x \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) \cdot \left(x \cdot {x}^{2}\right)} + x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x \cdot {x}^{2}, x\right)} \]

   8. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x \cdot {x}^{2}, x\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{15} \cdot \color{blue}{\left(x \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), x \cdot {x}^{2}, x\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{2}{15} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), x \cdot {x}^{2}, x\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{2}{15} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), x \cdot {x}^{2}, x\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{2}{15} \cdot x\right) + \color{blue}{\frac{-1}{3}}, x \cdot {x}^{2}, x\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, \frac{2}{15} \cdot x, \frac{-1}{3}\right)}, x \cdot {x}^{2}, x\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{15}}, \frac{-1}{3}\right), x \cdot {x}^{2}, x\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{15}}, \frac{-1}{3}\right), x \cdot {x}^{2}, x\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right), \color{blue}{x \cdot {x}^{2}}, x\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   18. lower-*.f64 97.8

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

5. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot \frac{2}{15}\right)} + \frac{-1}{3}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + x \]

   2. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)} \cdot \left(x \cdot \left(x \cdot x\right)\right) + x \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) + x \]

   4. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + x \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + x \]

   6. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot x\right) \cdot \left(x \cdot x\right)} + x \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} + x \]

   8. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot x\right) \cdot x\right) \cdot x} + x \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot x\right) \cdot x, x, x\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot x\right) \cdot x}, x, x\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right)} \cdot x, x, x\right) \]

   12. lower-*.f64 97.8

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right)\right)} \cdot x, x, x\right) \]

7. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right)\right) \cdot x, x, x\right)} \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)}, x, x\right) \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)}, x, x\right) \]

   2. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right), x, x\right) \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right), x, x\right) \]

   4. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)\right)}, x, x\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}\right), x, x\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\color{blue}{{x}^{2} \cdot \frac{2}{15}} + \frac{-1}{3}\right), x, x\right) \]

   7. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{15}, \frac{-1}{3}\right)}, x, x\right) \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{15}, \frac{-1}{3}\right), x, x\right) \]

   9. lower-*.f64 97.8

      \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.13333333333333333, -0.3333333333333333\right), x, x\right) \]

10. Applied rewrites 97.8%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right)}, x, x\right) \]
11. Add Preprocessing

Alternative 6: 97.3% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (fma x (* x (fma x (* x 0.13333333333333333) -0.3333333333333333)) 1.0)))

C

double code(double x) {
	return x * fma(x, (x * fma(x, (x * 0.13333333333333333), -0.3333333333333333)), 1.0);
}

Julia

function code(x)
	return Float64(x * fma(x, Float64(x * fma(x, Float64(x * 0.13333333333333333), -0.3333333333333333)), 1.0))
end

Wolfram

code[x_] := N[(x * N[(x * N[(x * N[(x * N[(x * 0.13333333333333333), $MachinePrecision] + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.3%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) \cdot x + 1 \cdot x} \]

   3. *-lft-identity N/A

      \[\leadsto \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) \cdot x + \color{blue}{x} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} + x \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) \cdot \left(x \cdot {x}^{2}\right)} + x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x \cdot {x}^{2}, x\right)} \]

   8. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x \cdot {x}^{2}, x\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{15} \cdot \color{blue}{\left(x \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), x \cdot {x}^{2}, x\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{2}{15} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), x \cdot {x}^{2}, x\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{2}{15} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), x \cdot {x}^{2}, x\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{2}{15} \cdot x\right) + \color{blue}{\frac{-1}{3}}, x \cdot {x}^{2}, x\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, \frac{2}{15} \cdot x, \frac{-1}{3}\right)}, x \cdot {x}^{2}, x\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{15}}, \frac{-1}{3}\right), x \cdot {x}^{2}, x\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{15}}, \frac{-1}{3}\right), x \cdot {x}^{2}, x\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right), \color{blue}{x \cdot {x}^{2}}, x\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   18. lower-*.f64 97.8

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

5. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot \frac{2}{15}\right)} + \frac{-1}{3}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + x \]

   2. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)} \cdot \left(x \cdot \left(x \cdot x\right)\right) + x \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) + x \]

   4. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + x \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + x \]

   6. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot x\right) \cdot \left(x \cdot x\right)} + x \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} + x \]

   8. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot x\right) \cdot x\right) \cdot x} + x \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot x\right) \cdot x, x, x\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot x\right) \cdot x}, x, x\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right)} \cdot x, x, x\right) \]

   12. lower-*.f64 97.8

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right)\right)} \cdot x, x, x\right) \]

7. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right)\right) \cdot x, x, x\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{2}{15}\right)} + \frac{-1}{3}\right)\right) \cdot x\right) \cdot x + x \]

   2. lift-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)}\right) \cdot x\right) \cdot x + x \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right)} \cdot x\right) \cdot x + x \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right) \cdot x\right)} \cdot x + x \]

   5. distribute-lft1-in N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right) \cdot x + 1\right) \cdot x} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right) \cdot x + 1\right) \cdot x} \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right) \cdot x} + 1\right) \cdot x \]

   8. *-commutative N/A

      \[\leadsto \left(\color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right)} + 1\right) \cdot x \]

   9. lower-fma.f64 97.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), 1\right)} \cdot x \]

9. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), 1\right) \cdot x} \]

10. Final simplification 97.8%

    \[\leadsto x \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), 1\right) \]
11. Add Preprocessing

Alternative 7: 97.0% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.3333333333333333, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x (* (* x x) -0.3333333333333333) x))

C

double code(double x) {
	return fma(x, ((x * x) * -0.3333333333333333), x);
}

Julia

function code(x)
	return fma(x, Float64(Float64(x * x) * -0.3333333333333333), x)
end

Wolfram

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

Derivation

1. Initial program 10.3%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{-1}{3} \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   7. lower-*.f64 97.3

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

5. Applied rewrites 97.3%

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

6. Final simplification 97.3%

   \[\leadsto \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot -0.3333333333333333, x\right) \]
7. Add Preprocessing

Alternative 8: 97.0% accurate, 24.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \mathsf{fma}\left(-0.3333333333333333, x \cdot x, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (fma -0.3333333333333333 (* x x) 1.0)))

C

double code(double x) {
	return x * fma(-0.3333333333333333, (x * x), 1.0);
}

Julia

function code(x)
	return Float64(x * fma(-0.3333333333333333, Float64(x * x), 1.0))
end

Wolfram

code[x_] := N[(x * N[(-0.3333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 10.3%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, \frac{-1}{3} \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   7. lower-*.f64 97.3

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

5. Applied rewrites 97.3%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. distribute-lft1-in N/A

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

   5. lower-*.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. lower-fma.f64 97.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, x \cdot x, 1\right)} \cdot x \]

7. Applied rewrites 97.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.3333333333333333, x \cdot x, 1\right) \cdot x} \]

8. Final simplification 97.3%

   \[\leadsto x \cdot \mathsf{fma}\left(-0.3333333333333333, x \cdot x, 1\right) \]
9. Add Preprocessing

Alternative 9: 96.8% accurate, 422.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 10.3%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) \cdot x + 1 \cdot x} \]

   3. *-lft-identity N/A

      \[\leadsto \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) \cdot x + \color{blue}{x} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} + x \]

   5. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) \cdot \left(x \cdot {x}^{2}\right)} + x \]

   7. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x \cdot {x}^{2}, x\right)} \]

   8. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right)}, x \cdot {x}^{2}, x\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{2}{15} \cdot \color{blue}{\left(x \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), x \cdot {x}^{2}, x\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\frac{2}{15} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), x \cdot {x}^{2}, x\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot \left(\frac{2}{15} \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right), x \cdot {x}^{2}, x\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(x \cdot \left(\frac{2}{15} \cdot x\right) + \color{blue}{\frac{-1}{3}}, x \cdot {x}^{2}, x\right) \]

   13. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, \frac{2}{15} \cdot x, \frac{-1}{3}\right)}, x \cdot {x}^{2}, x\right) \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{15}}, \frac{-1}{3}\right), x \cdot {x}^{2}, x\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{2}{15}}, \frac{-1}{3}\right), x \cdot {x}^{2}, x\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right), \color{blue}{x \cdot {x}^{2}}, x\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   18. lower-*.f64 97.8

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

5. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), x \cdot \left(x \cdot x\right), x\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(x \cdot \color{blue}{\left(x \cdot \frac{2}{15}\right)} + \frac{-1}{3}\right) \cdot \left(x \cdot \left(x \cdot x\right)\right) + x \]

   2. lift-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)} \cdot \left(x \cdot \left(x \cdot x\right)\right) + x \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) + x \]

   4. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + x \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} + x \]

   6. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot x\right) \cdot \left(x \cdot x\right)} + x \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot x\right) \cdot \color{blue}{\left(x \cdot x\right)} + x \]

   8. associate-*r* N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot x\right) \cdot x\right) \cdot x} + x \]

   9. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot x\right) \cdot x, x, x\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right) \cdot x\right) \cdot x}, x, x\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right)} \cdot x, x, x\right) \]

   12. lower-*.f64 97.8

       \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right)\right)} \cdot x, x, x\right) \]

7. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\left(x \cdot \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right)\right) \cdot x, x, x\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left(\left(x \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{2}{15}\right)} + \frac{-1}{3}\right)\right) \cdot x\right) \cdot x + x \]

   2. lift-fma.f64 N/A

      \[\leadsto \left(\left(x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)}\right) \cdot x\right) \cdot x + x \]

   3. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right)} \cdot x\right) \cdot x + x \]

   4. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right) \cdot x\right)} \cdot x + x \]

   5. distribute-lft1-in N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right) \cdot x + 1\right) \cdot x} \]

   6. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right) \cdot x + 1\right) \cdot x} \]

   7. lift-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right) \cdot x} + 1\right) \cdot x \]

   8. *-commutative N/A

      \[\leadsto \left(\color{blue}{x \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot \frac{2}{15}, \frac{-1}{3}\right)\right)} + 1\right) \cdot x \]

   9. lower-fma.f64 97.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), 1\right)} \cdot x \]

9. Applied rewrites 97.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.13333333333333333, -0.3333333333333333\right), 1\right) \cdot x} \]

10. Taylor expanded in x around 0

    \[\leadsto \color{blue}{1} \cdot x \]
11. Step-by-step derivation

12. Applied rewrites 96.6%

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

13. Final simplification 96.6%

    \[\leadsto x \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))