Hyperbolic tangent

Percentage Accurate: 9.2% → 100.0%

Time: 12.0s

Alternatives: 6

Speedup: 422.0×

• could not determine a ground truth

  1. x = 5.457614829833846e+277

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: 9.2% 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 7.6%

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

   1. tanh-undef N/A

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

   2. tanh-lowering-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, 9.4× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0021164021164021165, -0.022222222222222223\right), 0.3333333333333333\right), 1\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  x
  (fma
   (* x x)
   (fma
    x
    (* x (fma x (* x 0.0021164021164021165) -0.022222222222222223))
    0.3333333333333333)
   1.0)))

C

double code(double x) {
	return x / fma((x * x), fma(x, (x * fma(x, (x * 0.0021164021164021165), -0.022222222222222223)), 0.3333333333333333), 1.0);
}

Julia

function code(x)
	return Float64(x / fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * 0.0021164021164021165), -0.022222222222222223)), 0.3333333333333333), 1.0))
end

Wolfram

code[x_] := N[(x / N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.0021164021164021165), $MachinePrecision] + -0.022222222222222223), $MachinePrecision]), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.6%

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

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{e^{x} - e^{\mathsf{neg}\left(x\right)}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{e^{x} - e^{\mathsf{neg}\left(x\right)}}}} \]

   3. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{e^{x} - e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{e^{x} - e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}}}} \]

   5. tanh-undef N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\tanh x}}} \]

   6. tanh-lowering-tanh.f64 99.7

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\tanh x}}} \]

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\tanh x}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{2}{945} \cdot {x}^{2} - \frac{1}{45}\right)\right)}{x}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{2}{945} \cdot {x}^{2} - \frac{1}{45}\right)\right)}{x}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{2}{945} \cdot {x}^{2} - \frac{1}{45}\right)\right) + 1}}{x}} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{2}{945} \cdot {x}^{2} - \frac{1}{45}\right), 1\right)}}{x}} \]

   4. unpow2 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{2}{945} \cdot {x}^{2} - \frac{1}{45}\right), 1\right)}{x}} \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} + {x}^{2} \cdot \left(\frac{2}{945} \cdot {x}^{2} - \frac{1}{45}\right), 1\right)}{x}} \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{2}{945} \cdot {x}^{2} - \frac{1}{45}\right) + \frac{1}{3}}, 1\right)}{x}} \]

   7. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{945} \cdot {x}^{2} - \frac{1}{45}, \frac{1}{3}\right)}, 1\right)}{x}} \]

   8. unpow2 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{945} \cdot {x}^{2} - \frac{1}{45}, \frac{1}{3}\right), 1\right)}{x}} \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{945} \cdot {x}^{2} - \frac{1}{45}, \frac{1}{3}\right), 1\right)}{x}} \]

   10. sub-neg N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{2}{945} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{45}\right)\right)}, \frac{1}{3}\right), 1\right)}{x}} \]

   11. *-commutative N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{2}{945}} + \left(\mathsf{neg}\left(\frac{1}{45}\right)\right), \frac{1}{3}\right), 1\right)}{x}} \]

   12. metadata-eval N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, {x}^{2} \cdot \frac{2}{945} + \color{blue}{\frac{-1}{45}}, \frac{1}{3}\right), 1\right)}{x}} \]

   13. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{2}{945}, \frac{-1}{45}\right)}, \frac{1}{3}\right), 1\right)}{x}} \]

   14. unpow2 N/A

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{2}{945}, \frac{-1}{45}\right), \frac{1}{3}\right), 1\right)}{x}} \]

   15. *-lowering-*.f64 98.6

       \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.0021164021164021165, -0.022222222222222223\right), 0.3333333333333333\right), 1\right)}{x}} \]

7. Simplified 98.6%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0021164021164021165, -0.022222222222222223\right), 0.3333333333333333\right), 1\right)}{x}}} \]
8. Step-by-step derivation

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{x}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{945} + \frac{-1}{45}\right) + \frac{1}{3}\right) + 1}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{945} + \frac{-1}{45}\right) + \frac{1}{3}\right) + 1}} \]

   3. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{x}{\color{blue}{\mathsf{fma}\left(x \cdot x, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{945} + \frac{-1}{45}\right) + \frac{1}{3}, 1\right)}} \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(\color{blue}{x \cdot x}, \left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{945} + \frac{-1}{45}\right) + \frac{1}{3}, 1\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{945} + \frac{-1}{45}\right)\right)} + \frac{1}{3}, 1\right)} \]

   6. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{945} + \frac{-1}{45}\right), \frac{1}{3}\right)}, 1\right)} \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\left(x \cdot x\right) \cdot \frac{2}{945} + \frac{-1}{45}\right)}, \frac{1}{3}\right), 1\right)} \]

   8. associate-*l* N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{2}{945}\right)} + \frac{-1}{45}\right), \frac{1}{3}\right), 1\right)} \]

   9. accelerator-lowering-fma.f64 N/A

      \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{2}{945}, \frac{-1}{45}\right)}, \frac{1}{3}\right), 1\right)} \]

   10. *-lowering-*.f64 98.8

       \[\leadsto \frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 0.0021164021164021165}, -0.022222222222222223\right), 0.3333333333333333\right), 1\right)} \]

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.0021164021164021165, -0.022222222222222223\right), 0.3333333333333333\right), 1\right)}} \]
10. Add Preprocessing

Alternative 3: 97.0% accurate, 15.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.6%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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. accelerator-lowering-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. accelerator-lowering-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. *-lowering-*.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. *-lowering-*.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. *-lowering-*.f64 98.8

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

5. Simplified 98.8%

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

Alternative 4: 97.0% accurate, 18.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.6%

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

   1. clear-num N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{e^{x} - e^{\mathsf{neg}\left(x\right)}}}} \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} + e^{\mathsf{neg}\left(x\right)}}{e^{x} - e^{\mathsf{neg}\left(x\right)}}}} \]

   3. clear-num N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{e^{x} - e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}}}} \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{e^{x} - e^{\mathsf{neg}\left(x\right)}}{e^{x} + e^{\mathsf{neg}\left(x\right)}}}}} \]

   5. tanh-undef N/A

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\tanh x}}} \]

   6. tanh-lowering-tanh.f64 99.7

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\tanh x}}} \]

4. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\tanh x}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{1 + \frac{1}{3} \cdot {x}^{2}}{x}}} \]
6. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1 + \frac{1}{3} \cdot {x}^{2}}{x}}} \]

   2. +-commutative N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{\frac{1}{3} \cdot {x}^{2} + 1}}{x}} \]

   3. *-commutative N/A

      \[\leadsto \frac{1}{\frac{\color{blue}{{x}^{2} \cdot \frac{1}{3}} + 1}{x}} \]

   4. accelerator-lowering-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. *-lowering-*.f64 98.5

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

7. Simplified 98.5%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, 0.3333333333333333, 1\right)}{x}}} \]
8. Step-by-step derivation

   1. clear-num N/A

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

   2. /-lowering-/.f64 N/A

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

   3. associate-*l* N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. *-lowering-*.f64 98.8

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

9. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\frac{x}{\mathsf{fma}\left(x, x \cdot 0.3333333333333333, 1\right)}} \]
10. 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 7.6%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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. accelerator-lowering-fma.f64 N/A

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

   5. *-lowering-*.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. *-lowering-*.f64 98.6

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

5. Simplified 98.6%

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

6. Final simplification 98.6%

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

Alternative 6: 96.4% 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 7.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 98.4%

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

Reproduce

herbie shell --seed 2024205 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))