Hyperbolic tangent

Percentage Accurate: 9.3% → 100.0%

Time: 11.1s

Alternatives: 5

Speedup: 70.3×

• could not determine a ground truth

  1. x = 9.26901258269821e+241

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: 9.3% 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 7.1%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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 egg-rr 100.0%

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

Alternative 2: 97.1% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\left(x \cdot \left(x \cdot x\right)\right) \cdot -0.1111111111111111, \frac{-1}{\mathsf{fma}\left(x \cdot x, -0.13333333333333333, -0.3333333333333333\right)}, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (* (* x (* x x)) -0.1111111111111111)
  (/ -1.0 (fma (* x x) -0.13333333333333333 -0.3333333333333333))
  x))

C

double code(double x) {
	return fma(((x * (x * x)) * -0.1111111111111111), (-1.0 / fma((x * x), -0.13333333333333333, -0.3333333333333333)), x);
}

Julia

function code(x)
	return fma(Float64(Float64(x * Float64(x * x)) * -0.1111111111111111), Float64(-1.0 / fma(Float64(x * x), -0.13333333333333333, -0.3333333333333333)), x)
end

Wolfram

code[x_] := N[(N[(N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] * -0.1111111111111111), $MachinePrecision] * N[(-1.0 / N[(N[(x * x), $MachinePrecision] * -0.13333333333333333 + -0.3333333333333333), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 7.1%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. +-commutative N/A

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

   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. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 99.2

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

5. Simplified 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. flip-+ N/A

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

   7. associate-*l/ N/A

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

   8. div-inv N/A

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

   9. lower-fma.f64 N/A

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

7. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.017777777777777778, -0.1111111111111111\right) \cdot \left(x \cdot \left(x \cdot x\right)\right), \frac{-1}{\mathsf{fma}\left(x \cdot x, -0.13333333333333333, -0.3333333333333333\right)}, x\right)} \]

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. cube-mult N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.3

      \[\leadsto \mathsf{fma}\left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot -0.1111111111111111, \frac{-1}{\mathsf{fma}\left(x \cdot x, -0.13333333333333333, -0.3333333333333333\right)}, x\right) \]

10. Simplified 99.3%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot -0.1111111111111111}, \frac{-1}{\mathsf{fma}\left(x \cdot x, -0.13333333333333333, -0.3333333333333333\right)}, x\right) \]
11. Add Preprocessing

Alternative 3: 96.9% accurate, 15.1× speedup

Math

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

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(x, Float64(x * 0.13333333333333333), -0.3333333333333333)), x, x)
end

Wolfram

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

Derivation

1. Initial program 7.1%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. +-commutative N/A

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

   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. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 99.2

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

5. Simplified 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. lower-*.f64 99.2

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

   13. lift-fma.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. associate-*l* N/A

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

   16. lower-fma.f64 N/A

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

   17. lower-*.f64 99.2

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 4: 96.5% 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 7.1%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 99.1

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

5. Simplified 99.1%

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

Alternative 5: 16.5% accurate, 70.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.16666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.16666666666666666))

C

double code(double x) {
	return x * 0.16666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.16666666666666666d0
end function

Java

public static double code(double x) {
	return x * 0.16666666666666666;
}

Python

def code(x):
	return x * 0.16666666666666666

Julia

function code(x)
	return Float64(x * 0.16666666666666666)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.16666666666666666;
end

Wolfram

code[x_] := N[(x * 0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 7.1%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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. distribute-rgt-in N/A

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

   2. *-lft-identity N/A

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

   3. +-commutative N/A

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

   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. *-commutative N/A

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

   10. metadata-eval N/A

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

   11. lower-fma.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 99.2

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

5. Simplified 99.2%

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

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. lift-fma.f64 N/A

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

   6. flip-+ N/A

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

   7. associate-*l/ N/A

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

   8. div-inv N/A

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

   9. lower-fma.f64 N/A

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

7. Applied egg-rr 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot 0.017777777777777778, -0.1111111111111111\right) \cdot \left(x \cdot \left(x \cdot x\right)\right), \frac{-1}{\mathsf{fma}\left(x \cdot x, -0.13333333333333333, -0.3333333333333333\right)}, x\right)} \]

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. cube-mult N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 99.3

      \[\leadsto \mathsf{fma}\left(\left(x \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot -0.1111111111111111, \frac{-1}{\mathsf{fma}\left(x \cdot x, -0.13333333333333333, -0.3333333333333333\right)}, x\right) \]

10. Simplified 99.3%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\left(x \cdot \left(x \cdot x\right)\right) \cdot -0.1111111111111111}, \frac{-1}{\mathsf{fma}\left(x \cdot x, -0.13333333333333333, -0.3333333333333333\right)}, x\right) \]

11. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\frac{1}{6} \cdot x} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \color{blue}{x \cdot \frac{1}{6}} \]

    2. lower-*.f64 16.6

       \[\leadsto \color{blue}{x \cdot 0.16666666666666666} \]

13. Simplified 16.6%

    \[\leadsto \color{blue}{x \cdot 0.16666666666666666} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))