Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 10.5s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \frac{1 - t_0}{1 + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x)))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) * tan(x)
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Initial Program: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan x \cdot \tan x\\ \frac{1 - t_0}{1 + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (tan x) (tan x)))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = tan(x) * tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) * tan(x)
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.tan(x) * Math.tan(x);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.tan(x) * math.tan(x)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = Float64(tan(x) * tan(x))
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) * tan(x);
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Tan[x], $MachinePrecision] * N[Tan[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + {\tan x}^{2}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fma (tan x) (- (tan x)) 1.0) (+ 1.0 (pow (tan x) 2.0))))

C

double code(double x) {
	return fma(tan(x), -tan(x), 1.0) / (1.0 + pow(tan(x), 2.0));
}

Julia

function code(x)
	return Float64(fma(tan(x), Float64(-tan(x)), 1.0) / Float64(1.0 + (tan(x) ^ 2.0)))
end

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Step-by-step derivation

   1. sub-neg 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

   4. fma-def 99.5%

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

3. Applied egg-rr 99.5%

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

   1. add-log-exp 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. log-prod 98.0%

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

   4. metadata-eval 98.0%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)} \]

   5. add-log-exp 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{\tan x \cdot \tan x}\right)} \]

   6. pow2 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{{\tan x}^{2}}\right)} \]

5. Applied egg-rr 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \color{blue}{\left(0 + {\tan x}^{2}\right)}} \]
6. Step-by-step derivation

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 2: 99.5% accurate, 0.8× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (- 1.0 (* (tan x) (tan x))) (fma (tan x) (tan x) 1.0)))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Step-by-step derivation

   1. +-commutative 99.5%

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

   2. fma-def 99.5%

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

3. Simplified 99.5%

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

4. Final simplification 99.5%

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

Alternative 3: 59.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{-1 - t_0}{t_0 + -1} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))) (/ (- -1.0 t_0) (+ t_0 -1.0))))

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	return (-1.0 - t_0) / (t_0 + -1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) ** 2.0d0
    code = ((-1.0d0) - t_0) / (t_0 + (-1.0d0))
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.tan(x), 2.0);
	return (-1.0 - t_0) / (t_0 + -1.0);
}

Python

def code(x):
	t_0 = math.pow(math.tan(x), 2.0)
	return (-1.0 - t_0) / (t_0 + -1.0)

Julia

function code(x)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(-1.0 - t_0) / Float64(t_0 + -1.0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (-1.0 - t_0) / (t_0 + -1.0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(-1.0 - t$95$0), $MachinePrecision] / N[(t$95$0 + -1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Step-by-step derivation

   1. +-commutative 99.5%

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

   2. fma-udef 99.5%

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

   3. div-sub 99.3%

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

   4. fma-udef 99.2%

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

   5. +-commutative 99.2%

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

   6. add-sqr-sqrt 99.1%

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

   7. pow2 99.1%

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

   8. hypot-1-def 99.1%

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

   9. pow2 99.1%

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

   10. fma-udef 99.1%

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

   11. +-commutative 99.1%

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

   12. add-sqr-sqrt 99.0%

       \[\leadsto \frac{1}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}} - \frac{{\tan x}^{2}}{\color{blue}{\sqrt{1 + \tan x \cdot \tan x} \cdot \sqrt{1 + \tan x \cdot \tan x}}} \]

   13. pow2 99.0%

       \[\leadsto \frac{1}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}} - \frac{{\tan x}^{2}}{\color{blue}{{\left(\sqrt{1 + \tan x \cdot \tan x}\right)}^{2}}} \]

   14. hypot-1-def 99.1%

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

3. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{1}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}} - \frac{{\tan x}^{2}}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{2}}} \]
4. Step-by-step derivation

   1. frac-2neg 99.1%

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

   2. metadata-eval 99.1%

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

   3. frac-2neg 99.1%

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

   4. sub-div 99.2%

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

   5. pow2 99.2%

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

   6. distribute-rgt-neg-in 99.2%

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

   7. add-sqr-sqrt 48.0%

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

   8. sqrt-unprod 72.7%

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

   9. sqr-neg 72.7%

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

   10. sqrt-prod 24.6%

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

   11. add-sqr-sqrt 49.1%

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

   12. pow2 49.1%

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

   13. unpow2 49.1%

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

5. Applied egg-rr 54.8%

   \[\leadsto \color{blue}{\frac{-1 - {\tan x}^{2}}{{\tan x}^{2} + -1}} \]

6. Final simplification 54.8%

   \[\leadsto \frac{-1 - {\tan x}^{2}}{{\tan x}^{2} + -1} \]

Alternative 4: 99.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\tan x}^{2}\\ \frac{1 - t_0}{1 + t_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (tan x) 2.0))) (/ (- 1.0 t_0) (+ 1.0 t_0))))

C

double code(double x) {
	double t_0 = pow(tan(x), 2.0);
	return (1.0 - t_0) / (1.0 + t_0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = tan(x) ** 2.0d0
    code = (1.0d0 - t_0) / (1.0d0 + t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.pow(Math.tan(x), 2.0);
	return (1.0 - t_0) / (1.0 + t_0);
}

Python

def code(x):
	t_0 = math.pow(math.tan(x), 2.0)
	return (1.0 - t_0) / (1.0 + t_0)

Julia

function code(x)
	t_0 = tan(x) ^ 2.0
	return Float64(Float64(1.0 - t_0) / Float64(1.0 + t_0))
end

MATLAB

function tmp = code(x)
	t_0 = tan(x) ^ 2.0;
	tmp = (1.0 - t_0) / (1.0 + t_0);
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]}, N[(N[(1.0 - t$95$0), $MachinePrecision] / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Step-by-step derivation

   1. sub-neg 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

   4. fma-def 99.5%

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

3. Applied egg-rr 99.5%

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

   1. add-log-exp 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. log-prod 98.0%

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

   4. metadata-eval 98.0%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)} \]

   5. add-log-exp 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{\tan x \cdot \tan x}\right)} \]

   6. pow2 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{{\tan x}^{2}}\right)} \]

5. Applied egg-rr 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \color{blue}{\left(0 + {\tan x}^{2}\right)}} \]
6. Step-by-step derivation

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \color{blue}{{\tan x}^{2}}} \]
8. Step-by-step derivation

   1. fma-udef 99.5%

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

   2. distribute-rgt-neg-in 99.5%

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

   3. pow2 99.5%

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

   4. +-commutative 99.5%

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

   5. sub-neg 99.5%

      \[\leadsto \frac{\color{blue}{1 - {\tan x}^{2}}}{1 + {\tan x}^{2}} \]

9. Applied egg-rr 99.5%

   \[\leadsto \frac{\color{blue}{1 - {\tan x}^{2}}}{1 + {\tan x}^{2}} \]

10. Final simplification 99.5%

    \[\leadsto \frac{1 - {\tan x}^{2}}{1 + {\tan x}^{2}} \]

Alternative 5: 55.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {\tan x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ 1.0 (pow (tan x) 2.0))))

C

double code(double x) {
	return 1.0 / (1.0 + pow(tan(x), 2.0));
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 / (1.0 + Math.pow(Math.tan(x), 2.0));
}

Python

def code(x):
	return 1.0 / (1.0 + math.pow(math.tan(x), 2.0))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 + (tan(x) ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 + (tan(x) ^ 2.0));
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 + N[Power[N[Tan[x], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Step-by-step derivation

   1. sub-neg 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

   4. fma-def 99.5%

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

3. Applied egg-rr 99.5%

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

   1. add-log-exp 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. log-prod 98.0%

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

   4. metadata-eval 98.0%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)} \]

   5. add-log-exp 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{\tan x \cdot \tan x}\right)} \]

   6. pow2 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{{\tan x}^{2}}\right)} \]

5. Applied egg-rr 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \color{blue}{\left(0 + {\tan x}^{2}\right)}} \]
6. Step-by-step derivation

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0 49.4%

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

9. Final simplification 49.4%

   \[\leadsto \frac{1}{1 + {\tan x}^{2}} \]

Alternative 6: 58.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 - {\tan x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (- 1.0 (pow (tan x) 2.0))))

C

double code(double x) {
	return 1.0 / (1.0 - pow(tan(x), 2.0));
}

Fortran

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

Java

public static double code(double x) {
	return 1.0 / (1.0 - Math.pow(Math.tan(x), 2.0));
}

Python

def code(x):
	return 1.0 / (1.0 - math.pow(math.tan(x), 2.0))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 - (tan(x) ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 - (tan(x) ^ 2.0));
end

Wolfram

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

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Step-by-step derivation

   1. sub-neg 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

   4. fma-def 99.5%

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

3. Applied egg-rr 99.5%

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

   1. add-log-exp 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. log-prod 98.0%

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

   4. metadata-eval 98.0%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)} \]

   5. add-log-exp 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{\tan x \cdot \tan x}\right)} \]

   6. pow2 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{{\tan x}^{2}}\right)} \]

5. Applied egg-rr 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \color{blue}{\left(0 + {\tan x}^{2}\right)}} \]
6. Step-by-step derivation

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0 49.4%

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

   1. +-commutative 49.4%

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

   2. pow2 49.4%

      \[\leadsto \frac{1}{\color{blue}{\tan x \cdot \tan x} + 1} \]

   3. fma-def 49.4%

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

   4. add-sqr-sqrt 24.8%

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

   5. sqrt-prod 51.3%

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

   6. sqr-neg 51.3%

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

   7. sqrt-unprod 26.5%

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

   8. add-sqr-sqrt 53.6%

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

   9. fma-udef 53.6%

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

   10. distribute-rgt-neg-in 53.6%

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

   11. pow2 53.6%

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

   12. +-commutative 53.6%

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

   13. sub-neg 53.6%

       \[\leadsto \frac{1}{\color{blue}{1 - {\tan x}^{2}}} \]

10. Applied egg-rr 53.6%

    \[\leadsto \frac{1}{\color{blue}{1 - {\tan x}^{2}}} \]

11. Final simplification 53.6%

    \[\leadsto \frac{1}{1 - {\tan x}^{2}} \]

Alternative 7: 55.4% accurate, 411.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 99.5%

   \[\frac{1 - \tan x \cdot \tan x}{1 + \tan x \cdot \tan x} \]
2. Step-by-step derivation

   1. sub-neg 99.5%

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

   2. +-commutative 99.5%

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

   3. distribute-rgt-neg-in 99.5%

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

   4. fma-def 99.5%

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

3. Applied egg-rr 99.5%

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

   1. add-log-exp 98.0%

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

   2. *-un-lft-identity 98.0%

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

   3. log-prod 98.0%

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

   4. metadata-eval 98.0%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(\color{blue}{0} + \log \left(e^{\tan x \cdot \tan x}\right)\right)} \]

   5. add-log-exp 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{\tan x \cdot \tan x}\right)} \]

   6. pow2 99.5%

      \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \left(0 + \color{blue}{{\tan x}^{2}}\right)} \]

5. Applied egg-rr 99.5%

   \[\leadsto \frac{\mathsf{fma}\left(\tan x, -\tan x, 1\right)}{1 + \color{blue}{\left(0 + {\tan x}^{2}\right)}} \]
6. Step-by-step derivation

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Taylor expanded in x around 0 49.1%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 49.1%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))