tanhf (example 3.4)

Percentage Accurate: 53.0% → 100.0%

Time: 8.9s

Alternatives: 3

Speedup: 2.0×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{\sin x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (sin x)))

C

double code(double x) {
	return (1.0 - cos(x)) / sin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / sin(x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / Math.sin(x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / math.sin(x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / sin(x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / sin(x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]

Initial Program: 53.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{\sin x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (sin x)))

C

double code(double x) {
	return (1.0 - cos(x)) / sin(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / sin(x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / Math.sin(x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / math.sin(x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / sin(x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / sin(x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[Sin[x], $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan \left(\frac{x}{2}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (tan (/ x 2.0)))

C

double code(double x) {
	return tan((x / 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x / 2.0d0))
end function

Java

public static double code(double x) {
	return Math.tan((x / 2.0));
}

Python

def code(x):
	return math.tan((x / 2.0))

Julia

function code(x)
	return tan(Float64(x / 2.0))
end

MATLAB

function tmp = code(x)
	tmp = tan((x / 2.0));
end

Wolfram

code[x_] := N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{1 - \cos x}{\sin x} \]
2. Step-by-step derivation

   1. hang-p0-tan 100.0%

      \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right)} \]

4. Final simplification 100.0%

   \[\leadsto \tan \left(\frac{x}{2}\right) \]

Alternative 2: 51.4% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{--2}{x} - x \cdot 0.16666666666666666} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 1.0 (- (/ (- -2.0) x) (* x 0.16666666666666666))))

C

double code(double x) {
	return 1.0 / ((-(-2.0) / x) - (x * 0.16666666666666666));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / ((-(-2.0d0) / x) - (x * 0.16666666666666666d0))
end function

Java

public static double code(double x) {
	return 1.0 / ((-(-2.0) / x) - (x * 0.16666666666666666));
}

Python

def code(x):
	return 1.0 / ((-(-2.0) / x) - (x * 0.16666666666666666))

Julia

function code(x)
	return Float64(1.0 / Float64(Float64(Float64(-(-2.0)) / x) - Float64(x * 0.16666666666666666)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / ((-(-2.0) / x) - (x * 0.16666666666666666));
end

Wolfram

code[x_] := N[(1.0 / N[(N[((--2.0) / x), $MachinePrecision] - N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{1 - \cos x}{\sin x} \]
2. Step-by-step derivation

   1. hang-p0-tan 100.0%

      \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right)} \]
4. Step-by-step derivation

   1. tan-quot 99.7%

      \[\leadsto \color{blue}{\frac{\sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)}} \]

   2. clear-num 99.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{\cos \left(\frac{x}{2}\right)}{\sin \left(\frac{x}{2}\right)}}} \]

   3. clear-num 99.5%

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{\frac{\sin \left(\frac{x}{2}\right)}{\cos \left(\frac{x}{2}\right)}}}} \]

   4. tan-quot 99.8%

      \[\leadsto \frac{1}{\frac{1}{\color{blue}{\tan \left(\frac{x}{2}\right)}}} \]

   5. div-inv 99.8%

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

   6. metadata-eval 99.8%

      \[\leadsto \frac{1}{\frac{1}{\tan \left(x \cdot \color{blue}{0.5}\right)}} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{1}{\tan \left(x \cdot 0.5\right)}}} \]

6. Taylor expanded in x around 0 50.2%

   \[\leadsto \frac{1}{\color{blue}{-0.16666666666666666 \cdot x + 2 \cdot \frac{1}{x}}} \]
7. Step-by-step derivation

   1. *-un-lft-identity 50.2%

      \[\leadsto \frac{1}{\color{blue}{1 \cdot \left(-0.16666666666666666 \cdot x + 2 \cdot \frac{1}{x}\right)}} \]

   2. metadata-eval 50.2%

      \[\leadsto \frac{1}{\color{blue}{\left(--1\right)} \cdot \left(-0.16666666666666666 \cdot x + 2 \cdot \frac{1}{x}\right)} \]

   3. metadata-eval 50.2%

      \[\leadsto \frac{1}{\left(-\color{blue}{\left(-1\right)}\right) \cdot \left(-0.16666666666666666 \cdot x + 2 \cdot \frac{1}{x}\right)} \]

   4. distribute-lft-neg-in 50.2%

      \[\leadsto \frac{1}{\color{blue}{-\left(-1\right) \cdot \left(-0.16666666666666666 \cdot x + 2 \cdot \frac{1}{x}\right)}} \]

   5. distribute-lft-neg-in 50.2%

      \[\leadsto \frac{1}{-\color{blue}{\left(-1 \cdot \left(-0.16666666666666666 \cdot x + 2 \cdot \frac{1}{x}\right)\right)}} \]

   6. distribute-rgt-neg-in 50.2%

      \[\leadsto \frac{1}{-\color{blue}{1 \cdot \left(-\left(-0.16666666666666666 \cdot x + 2 \cdot \frac{1}{x}\right)\right)}} \]

   7. *-un-lft-identity 50.2%

      \[\leadsto \frac{1}{-\color{blue}{\left(-\left(-0.16666666666666666 \cdot x + 2 \cdot \frac{1}{x}\right)\right)}} \]

   8. +-commutative 50.2%

      \[\leadsto \frac{1}{-\left(-\color{blue}{\left(2 \cdot \frac{1}{x} + -0.16666666666666666 \cdot x\right)}\right)} \]

   9. distribute-neg-in 50.2%

      \[\leadsto \frac{1}{-\color{blue}{\left(\left(-2 \cdot \frac{1}{x}\right) + \left(--0.16666666666666666 \cdot x\right)\right)}} \]

   10. div-inv 50.2%

       \[\leadsto \frac{1}{-\left(\left(-\color{blue}{\frac{2}{x}}\right) + \left(--0.16666666666666666 \cdot x\right)\right)} \]

   11. distribute-neg-frac 50.2%

       \[\leadsto \frac{1}{-\left(\color{blue}{\frac{-2}{x}} + \left(--0.16666666666666666 \cdot x\right)\right)} \]

   12. metadata-eval 50.2%

       \[\leadsto \frac{1}{-\left(\frac{\color{blue}{-2}}{x} + \left(--0.16666666666666666 \cdot x\right)\right)} \]

   13. *-commutative 50.2%

       \[\leadsto \frac{1}{-\left(\frac{-2}{x} + \left(-\color{blue}{x \cdot -0.16666666666666666}\right)\right)} \]

   14. distribute-rgt-neg-in 50.2%

       \[\leadsto \frac{1}{-\left(\frac{-2}{x} + \color{blue}{x \cdot \left(--0.16666666666666666\right)}\right)} \]

   15. metadata-eval 50.2%

       \[\leadsto \frac{1}{-\left(\frac{-2}{x} + x \cdot \color{blue}{0.16666666666666666}\right)} \]

8. Applied egg-rr 50.2%

   \[\leadsto \frac{1}{\color{blue}{-\left(\frac{-2}{x} + x \cdot 0.16666666666666666\right)}} \]

9. Final simplification 50.2%

   \[\leadsto \frac{1}{\frac{--2}{x} - x \cdot 0.16666666666666666} \]

Alternative 3: 50.9% accurate, 68.3× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.5))

C

double code(double x) {
	return x * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.5d0
end function

Java

public static double code(double x) {
	return x * 0.5;
}

Python

def code(x):
	return x * 0.5

Julia

function code(x)
	return Float64(x * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.5;
end

Wolfram

code[x_] := N[(x * 0.5), $MachinePrecision]

Derivation

1. Initial program 54.2%

   \[\frac{1 - \cos x}{\sin x} \]
2. Step-by-step derivation

   1. hang-p0-tan 100.0%

      \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right)} \]

3. Simplified 100.0%

   \[\leadsto \color{blue}{\tan \left(\frac{x}{2}\right)} \]

4. Taylor expanded in x around 0 49.4%

   \[\leadsto \color{blue}{0.5 \cdot x} \]

5. Final simplification 49.4%

   \[\leadsto x \cdot 0.5 \]

Developer target: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \tan \left(\frac{x}{2}\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (tan (/ x 2.0)))

C

double code(double x) {
	return tan((x / 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = tan((x / 2.0d0))
end function

Java

public static double code(double x) {
	return Math.tan((x / 2.0));
}

Python

def code(x):
	return math.tan((x / 2.0))

Julia

function code(x)
	return tan(Float64(x / 2.0))
end

MATLAB

function tmp = code(x)
	tmp = tan((x / 2.0));
end

Wolfram

code[x_] := N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision]

Reproduce

herbie shell --seed 2023336 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64

  :herbie-target
  (tan (/ x 2.0))

  (/ (- 1.0 (cos x)) (sin x)))