tanhf (example 3.4)

Percentage Accurate: 52.6% → 100.0%

Time: 6.1s

Alternatives: 7

Speedup: 25.6×

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: 52.6% 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 56.3%

   \[\frac{1 - \cos x}{\sin x} \]
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. Add Preprocessing
5. Add Preprocessing

Alternative 2: 53.0% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-4 + \frac{16}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.1) (* x 0.5) (+ 1.0 (/ (+ -4.0 (/ 16.0 x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.1) {
		tmp = x * 0.5;
	} else {
		tmp = 1.0 + ((-4.0 + (16.0 / x)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.1d0) then
        tmp = x * 0.5d0
    else
        tmp = 1.0d0 + (((-4.0d0) + (16.0d0 / x)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.1) {
		tmp = x * 0.5;
	} else {
		tmp = 1.0 + ((-4.0 + (16.0 / x)) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.1:
		tmp = x * 0.5
	else:
		tmp = 1.0 + ((-4.0 + (16.0 / x)) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.1)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(1.0 + Float64(Float64(-4.0 + Float64(16.0 / x)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.1)
		tmp = x * 0.5;
	else
		tmp = 1.0 + ((-4.0 + (16.0 / x)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.1], N[(x * 0.5), $MachinePrecision], N[(1.0 + N[(N[(-4.0 + N[(16.0 / x), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.10000000000000009

   1. Initial program 40.6%

      \[\frac{1 - \cos x}{\sin x} \]
   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. Add Preprocessing

   5. Taylor expanded in x around 0 63.9%

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

   if 2.10000000000000009 < x

   1. Initial program 98.7%

      \[\frac{1 - \cos x}{\sin x} \]
   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. Add Preprocessing

   5. Taylor expanded in x around 0 2.5%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 2.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot x\right)\right)} \]

      2. expm1-undefine 2.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1} \]

      3. flip-- 2.3%

         \[\leadsto \color{blue}{\frac{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1}} \]

      4. log1p-undefine 2.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      5. rem-exp-log 2.3%

         \[\leadsto \frac{\color{blue}{\left(1 + 0.5 \cdot x\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      6. +-commutative 2.3%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot x + 1\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      7. log1p-undefine 2.3%

         \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      8. rem-exp-log 2.3%

         \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \color{blue}{\left(1 + 0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      9. +-commutative 2.3%

         \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \color{blue}{\left(0.5 \cdot x + 1\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      10. metadata-eval 2.3%

          \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \left(0.5 \cdot x + 1\right) - \color{blue}{1}}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      11. log1p-undefine 2.3%

          \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \left(0.5 \cdot x + 1\right) - 1}{e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} + 1} \]

      12. rem-exp-log 2.3%

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

      13. +-commutative 2.3%

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

   7. Applied egg-rr 2.3%

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

   8. Taylor expanded in x around 0 8.4%

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

   9. Taylor expanded in x around -inf 8.4%

      \[\leadsto \color{blue}{1 + -1 \cdot \frac{4 - 16 \cdot \frac{1}{x}}{x}} \]
   10. Step-by-step derivation

       1. associate-*r/ 8.4%

          \[\leadsto 1 + \color{blue}{\frac{-1 \cdot \left(4 - 16 \cdot \frac{1}{x}\right)}{x}} \]

       2. sub-neg 8.4%

          \[\leadsto 1 + \frac{-1 \cdot \color{blue}{\left(4 + \left(-16 \cdot \frac{1}{x}\right)\right)}}{x} \]

       3. distribute-rgt-in 8.4%

          \[\leadsto 1 + \frac{\color{blue}{4 \cdot -1 + \left(-16 \cdot \frac{1}{x}\right) \cdot -1}}{x} \]

       4. metadata-eval 8.4%

          \[\leadsto 1 + \frac{\color{blue}{-4} + \left(-16 \cdot \frac{1}{x}\right) \cdot -1}{x} \]

       5. distribute-rgt-neg-in 8.4%

          \[\leadsto 1 + \frac{-4 + \color{blue}{\left(16 \cdot \left(-\frac{1}{x}\right)\right)} \cdot -1}{x} \]

       6. distribute-frac-neg2 8.4%

          \[\leadsto 1 + \frac{-4 + \left(16 \cdot \color{blue}{\frac{1}{-x}}\right) \cdot -1}{x} \]

       7. associate-*r* 8.4%

          \[\leadsto 1 + \frac{-4 + \color{blue}{16 \cdot \left(\frac{1}{-x} \cdot -1\right)}}{x} \]

       8. *-commutative 8.4%

          \[\leadsto 1 + \frac{-4 + 16 \cdot \color{blue}{\left(-1 \cdot \frac{1}{-x}\right)}}{x} \]

       9. mul-1-neg 8.4%

          \[\leadsto 1 + \frac{-4 + 16 \cdot \color{blue}{\left(-\frac{1}{-x}\right)}}{x} \]

       10. distribute-neg-frac2 8.4%

           \[\leadsto 1 + \frac{-4 + 16 \cdot \color{blue}{\frac{1}{-\left(-x\right)}}}{x} \]

       11. remove-double-neg 8.4%

           \[\leadsto 1 + \frac{-4 + 16 \cdot \frac{1}{\color{blue}{x}}}{x} \]

       12. associate-*r/ 8.4%

           \[\leadsto 1 + \frac{-4 + \color{blue}{\frac{16 \cdot 1}{x}}}{x} \]

       13. metadata-eval 8.4%

           \[\leadsto 1 + \frac{-4 + \frac{\color{blue}{16}}{x}}{x} \]

   11. Simplified 8.4%

       \[\leadsto \color{blue}{1 + \frac{-4 + \frac{16}{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.1:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{-4 + \frac{16}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.0% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{4}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 3.1) (* x 0.5) (- 1.0 (/ 4.0 x))))

C

double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = x * 0.5;
	} else {
		tmp = 1.0 - (4.0 / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.1d0) then
        tmp = x * 0.5d0
    else
        tmp = 1.0d0 - (4.0d0 / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 3.1) {
		tmp = x * 0.5;
	} else {
		tmp = 1.0 - (4.0 / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.1:
		tmp = x * 0.5
	else:
		tmp = 1.0 - (4.0 / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.1)
		tmp = Float64(x * 0.5);
	else
		tmp = Float64(1.0 - Float64(4.0 / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.1)
		tmp = x * 0.5;
	else
		tmp = 1.0 - (4.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.1], N[(x * 0.5), $MachinePrecision], N[(1.0 - N[(4.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000009

   1. Initial program 40.6%

      \[\frac{1 - \cos x}{\sin x} \]
   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. Add Preprocessing

   5. Taylor expanded in x around 0 63.9%

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

   if 3.10000000000000009 < x

   1. Initial program 98.7%

      \[\frac{1 - \cos x}{\sin x} \]
   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. Add Preprocessing

   5. Taylor expanded in x around 0 2.5%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 2.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot x\right)\right)} \]

      2. expm1-undefine 2.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1} \]

      3. flip-- 2.3%

         \[\leadsto \color{blue}{\frac{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1}} \]

      4. log1p-undefine 2.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      5. rem-exp-log 2.3%

         \[\leadsto \frac{\color{blue}{\left(1 + 0.5 \cdot x\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      6. +-commutative 2.3%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot x + 1\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      7. log1p-undefine 2.3%

         \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      8. rem-exp-log 2.3%

         \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \color{blue}{\left(1 + 0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      9. +-commutative 2.3%

         \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \color{blue}{\left(0.5 \cdot x + 1\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      10. metadata-eval 2.3%

          \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \left(0.5 \cdot x + 1\right) - \color{blue}{1}}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      11. log1p-undefine 2.3%

          \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \left(0.5 \cdot x + 1\right) - 1}{e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} + 1} \]

      12. rem-exp-log 2.3%

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

      13. +-commutative 2.3%

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

   7. Applied egg-rr 2.3%

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

   8. Taylor expanded in x around 0 8.4%

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

   9. Taylor expanded in x around inf 8.4%

      \[\leadsto \color{blue}{1 - 4 \cdot \frac{1}{x}} \]
   10. Step-by-step derivation

       1. associate-*r/ 8.4%

          \[\leadsto 1 - \color{blue}{\frac{4 \cdot 1}{x}} \]

       2. metadata-eval 8.4%

          \[\leadsto 1 - \frac{\color{blue}{4}}{x} \]

   11. Simplified 8.4%

       \[\leadsto \color{blue}{1 - \frac{4}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 48.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{4}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.1% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \frac{x}{1 + \left(x \cdot 0.5 + 1\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ x (+ 1.0 (+ (* x 0.5) 1.0))))

C

double code(double x) {
	return x / (1.0 + ((x * 0.5) + 1.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x / (1.0d0 + ((x * 0.5d0) + 1.0d0))
end function

Java

public static double code(double x) {
	return x / (1.0 + ((x * 0.5) + 1.0));
}

Python

def code(x):
	return x / (1.0 + ((x * 0.5) + 1.0))

Julia

function code(x)
	return Float64(x / Float64(1.0 + Float64(Float64(x * 0.5) + 1.0)))
end

MATLAB

function tmp = code(x)
	tmp = x / (1.0 + ((x * 0.5) + 1.0));
end

Wolfram

code[x_] := N[(x / N[(1.0 + N[(N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.3%

   \[\frac{1 - \cos x}{\sin x} \]
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. Add Preprocessing

5. Taylor expanded in x around 0 47.3%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
6. Step-by-step derivation

   1. expm1-log1p-u 46.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot x\right)\right)} \]

   2. expm1-undefine 4.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1} \]

   3. flip-- 4.7%

      \[\leadsto \color{blue}{\frac{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1}} \]

   4. log1p-undefine 4.7%

      \[\leadsto \frac{e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

   5. rem-exp-log 4.7%

      \[\leadsto \frac{\color{blue}{\left(1 + 0.5 \cdot x\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

   6. +-commutative 4.7%

      \[\leadsto \frac{\color{blue}{\left(0.5 \cdot x + 1\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

   7. log1p-undefine 4.7%

      \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

   8. rem-exp-log 4.7%

      \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \color{blue}{\left(1 + 0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

   9. +-commutative 4.7%

      \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \color{blue}{\left(0.5 \cdot x + 1\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

   10. metadata-eval 4.7%

       \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \left(0.5 \cdot x + 1\right) - \color{blue}{1}}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

   11. log1p-undefine 4.7%

       \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \left(0.5 \cdot x + 1\right) - 1}{e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} + 1} \]

   12. rem-exp-log 5.4%

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

   13. +-commutative 5.4%

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

7. Applied egg-rr 5.4%

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

8. Taylor expanded in x around 0 50.2%

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

9. Final simplification 50.2%

   \[\leadsto \frac{x}{1 + \left(x \cdot 0.5 + 1\right)} \]
10. Add Preprocessing

Alternative 5: 53.0% accurate, 25.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.95:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.95) (* x 0.5) 2.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.95) {
		tmp = x * 0.5;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.95d0) then
        tmp = x * 0.5d0
    else
        tmp = 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.95) {
		tmp = x * 0.5;
	} else {
		tmp = 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.95:
		tmp = x * 0.5
	else:
		tmp = 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.95)
		tmp = Float64(x * 0.5);
	else
		tmp = 2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.95)
		tmp = x * 0.5;
	else
		tmp = 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.95], N[(x * 0.5), $MachinePrecision], 2.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.94999999999999996

   1. Initial program 40.6%

      \[\frac{1 - \cos x}{\sin x} \]
   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. Add Preprocessing

   5. Taylor expanded in x around 0 63.9%

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

   if 1.94999999999999996 < x

   1. Initial program 98.7%

      \[\frac{1 - \cos x}{\sin x} \]
   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. Add Preprocessing

   5. Taylor expanded in x around 0 2.5%

      \[\leadsto \color{blue}{0.5 \cdot x} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 2.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot x\right)\right)} \]

      2. expm1-undefine 2.5%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1} \]

      3. flip-- 2.3%

         \[\leadsto \color{blue}{\frac{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1}} \]

      4. log1p-undefine 2.3%

         \[\leadsto \frac{e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      5. rem-exp-log 2.3%

         \[\leadsto \frac{\color{blue}{\left(1 + 0.5 \cdot x\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      6. +-commutative 2.3%

         \[\leadsto \frac{\color{blue}{\left(0.5 \cdot x + 1\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      7. log1p-undefine 2.3%

         \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      8. rem-exp-log 2.3%

         \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \color{blue}{\left(1 + 0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      9. +-commutative 2.3%

         \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \color{blue}{\left(0.5 \cdot x + 1\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      10. metadata-eval 2.3%

          \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \left(0.5 \cdot x + 1\right) - \color{blue}{1}}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

      11. log1p-undefine 2.3%

          \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \left(0.5 \cdot x + 1\right) - 1}{e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} + 1} \]

      12. rem-exp-log 2.3%

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

      13. +-commutative 2.3%

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

   7. Applied egg-rr 2.3%

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

   8. Taylor expanded in x around 0 8.5%

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

   9. Taylor expanded in x around inf 8.5%

      \[\leadsto \color{blue}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.95:\\ \;\;\;\;x \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;2\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 6.8% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x) :precision binary64 2.0)

C

double code(double x) {
	return 2.0;
}

Fortran

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

Java

public static double code(double x) {
	return 2.0;
}

Python

def code(x):
	return 2.0

Julia

function code(x)
	return 2.0
end

MATLAB

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

Wolfram

code[x_] := 2.0

Derivation

1. Initial program 56.3%

   \[\frac{1 - \cos x}{\sin x} \]
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. Add Preprocessing

5. Taylor expanded in x around 0 47.3%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
6. Step-by-step derivation

   1. expm1-log1p-u 46.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot x\right)\right)} \]

   2. expm1-undefine 4.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1} \]

   3. flip-- 4.7%

      \[\leadsto \color{blue}{\frac{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1}} \]

   4. log1p-undefine 4.7%

      \[\leadsto \frac{e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

   5. rem-exp-log 4.7%

      \[\leadsto \frac{\color{blue}{\left(1 + 0.5 \cdot x\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

   6. +-commutative 4.7%

      \[\leadsto \frac{\color{blue}{\left(0.5 \cdot x + 1\right)} \cdot e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

   7. log1p-undefine 4.7%

      \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

   8. rem-exp-log 4.7%

      \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \color{blue}{\left(1 + 0.5 \cdot x\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

   9. +-commutative 4.7%

      \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \color{blue}{\left(0.5 \cdot x + 1\right)} - 1 \cdot 1}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

   10. metadata-eval 4.7%

       \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \left(0.5 \cdot x + 1\right) - \color{blue}{1}}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} + 1} \]

   11. log1p-undefine 4.7%

       \[\leadsto \frac{\left(0.5 \cdot x + 1\right) \cdot \left(0.5 \cdot x + 1\right) - 1}{e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} + 1} \]

   12. rem-exp-log 5.4%

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

   13. +-commutative 5.4%

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

7. Applied egg-rr 5.4%

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

8. Taylor expanded in x around 0 50.2%

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

9. Taylor expanded in x around inf 6.7%

   \[\leadsto \color{blue}{2} \]
10. Add Preprocessing

Alternative 7: 4.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 56.3%

   \[\frac{1 - \cos x}{\sin x} \]
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. Add Preprocessing

5. Taylor expanded in x around 0 47.3%

   \[\leadsto \color{blue}{0.5 \cdot x} \]
6. Step-by-step derivation

   1. expm1-log1p-u 46.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.5 \cdot x\right)\right)} \]

   2. expm1-undefine 4.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.5 \cdot x\right)} - 1} \]

   3. log1p-undefine 4.7%

      \[\leadsto e^{\color{blue}{\log \left(1 + 0.5 \cdot x\right)}} - 1 \]

   4. rem-exp-log 5.5%

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

   5. +-commutative 5.5%

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

7. Applied egg-rr 5.5%

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

8. Taylor expanded in x around 0 4.1%

   \[\leadsto \color{blue}{1} - 1 \]
9. Step-by-step derivation

   1. metadata-eval 4.1%

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 4.1%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Developer Target 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]

Reproduce

herbie shell --seed 2024149 
(FPCore (x)
  :name "tanhf (example 3.4)"
  :precision binary64

  :alt
  (! :herbie-platform default (tan (/ x 2)))

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