Hyperbolic tangent

Percentage Accurate: 9.6% → 100.0%

Time: 11.2s

Alternatives: 6

Speedup: 24.8×

• could not determine a ground truth

  1. x = 4.395840307774027e+126

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: 9.6% 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 9.3%

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

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

Alternative 2: 97.3% accurate, 9.6× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 9.3%

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

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. *-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 96.9

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

5. Simplified 96.9%

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

7. Applied egg-rr 96.9%

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

8. Taylor expanded in x around 0

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

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 97.1

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

10. Simplified 97.1%

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

Alternative 3: 97.1% accurate, 15.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.3%

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

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. *-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 96.9

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

5. Simplified 96.9%

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

Alternative 4: 97.1% accurate, 15.1× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 9.3%

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

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

   4. *-lft-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. sub-neg N/A

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

   11. *-commutative N/A

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

   12. unpow2 N/A

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

   13. associate-*l* N/A

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

   14. metadata-eval N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 96.9

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

5. Simplified 96.9%

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

   1. lift-*.f64 N/A

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

   2. lower-+.f64 N/A

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

   3. lower-*.f64 96.9

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

7. Applied egg-rr 96.9%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. metadata-eval N/A

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

   4. sub-neg N/A

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

   5. lower--.f64 96.9

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 96.9

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

9. Applied egg-rr 96.9%

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

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift--.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-*.f64 N/A

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

    7. associate-*l* N/A

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

    8. *-commutative N/A

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

    9. distribute-lft1-in N/A

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

    10. lower-*.f64 N/A

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

11. Applied egg-rr 96.9%

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

12. Final simplification 96.9%

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

Alternative 5: 96.6% 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 9.3%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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. *-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 96.5

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

5. Simplified 96.5%

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

Alternative 6: 5.0% accurate, 26.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return x * (x * (x * -0.3333333333333333));
}

Python

def code(x):
	return x * (x * (x * -0.3333333333333333))

Julia

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

MATLAB

function tmp = code(x)
	tmp = x * (x * (x * -0.3333333333333333));
end

Wolfram

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

Derivation

1. Initial program 9.3%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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. *-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 96.5

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

5. Simplified 96.5%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-1}{3} \cdot {x}^{3}} \]
7. Step-by-step derivation

   1. unpow3 N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 4.8

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

8. Simplified 4.8%

   \[\leadsto \color{blue}{x \cdot \left(x \cdot \left(x \cdot -0.3333333333333333\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))