Hyperbolic tangent

Percentage Accurate: 9.0% → 100.0%

Time: 10.0s

Alternatives: 4

Speedup: 24.8×

• could not determine a ground truth

  1. x = 1.2363482279990135e+294

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: 9.0% 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 9.7%

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

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{x} - e^{-x}}}{e^{x} + e^{-x}} \]

   3. lift-exp.f64 N/A

      \[\leadsto \frac{\color{blue}{e^{x}} - e^{-x}}{e^{x} + e^{-x}} \]

   4. lift-exp.f64 N/A

      \[\leadsto \frac{e^{x} - \color{blue}{e^{-x}}}{e^{x} + e^{-x}} \]

   5. lift-neg.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-exp.f64 N/A

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

   9. lift-neg.f64 N/A

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

   10. tanh-undef N/A

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

   11. 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: 97.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left({\left(\mathsf{fma}\left(-1.2, x \cdot x, -3\right)\right)}^{-1} \cdot x, x \cdot x, x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (* (pow (fma -1.2 (* x x) -3.0) -1.0) x) (* x x) x))

C

double code(double x) {
	return fma((pow(fma(-1.2, (x * x), -3.0), -1.0) * x), (x * x), x);
}

Julia

function code(x)
	return fma(Float64((fma(-1.2, Float64(x * x), -3.0) ^ -1.0) * x), Float64(x * x), x)
end

Wolfram

code[x_] := N[(N[(N[Power[N[(-1.2 * N[(x * x), $MachinePrecision] + -3.0), $MachinePrecision], -1.0], $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 9.7%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. pow-plus N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 95.8

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

5. Applied rewrites 95.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 95.8%

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

9. Applied rewrites 95.8%

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

10. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(\frac{1}{\frac{-6}{5} \cdot {x}^{2} - 3} \cdot x, x \cdot x, x\right) \]
11. Step-by-step derivation

12. Applied rewrites 96.1%

    \[\leadsto \mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(-1.2, x \cdot x, -3\right)} \cdot x, x \cdot x, x\right) \]

13. Final simplification 96.1%

    \[\leadsto \mathsf{fma}\left({\left(\mathsf{fma}\left(-1.2, x \cdot x, -3\right)\right)}^{-1} \cdot x, x \cdot x, x\right) \]
14. Add Preprocessing

Alternative 3: 97.1% accurate, 15.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 9.7%

   \[\frac{e^{x} - e^{-x}}{e^{x} + e^{-x}} \]
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-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. pow-plus N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. sub-neg N/A

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

   11. metadata-eval N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 95.8

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

5. Applied rewrites 95.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 95.8%

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

9. Applied rewrites 95.8%

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

Alternative 4: 96.7% accurate, 24.8× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 9.7%

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

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

   4. associate-*r* N/A

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

   5. *-rgt-identity N/A

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

   6. lower-fma.f64 N/A

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

   7. *-commutative N/A

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

   8. pow-plus N/A

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

   9. lower-pow.f64 N/A

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

   10. metadata-eval 95.2

       \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, -0.3333333333333333, x\right) \]

5. Applied rewrites 95.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 95.2%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (x)
  :name "Hyperbolic tangent"
  :precision binary64
  (/ (- (exp x) (exp (- x))) (+ (exp x) (exp (- x)))))