Trigonometry B

Percentage Accurate: 99.5% → 99.5%

Time: 11.5s

Alternatives: 8

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), 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. frac-2neg 99.5%

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

   2. div-inv 99.4%

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

   3. pow2 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-neg-in 99.4%

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

   6. neg-mul-1 99.4%

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

   7. metadata-eval 99.4%

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

   8. fma-def 99.4%

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

   9. pow2 99.4%

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

3. Applied egg-rr 99.4%

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

   1. associate-*r/ 99.5%

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

   2. *-rgt-identity 99.5%

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

   3. neg-sub0 99.5%

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

   4. associate--r- 99.5%

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

   5. metadata-eval 99.5%

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

   6. +-commutative 99.5%

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

   7. unpow2 99.5%

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

   8. fma-udef 99.5%

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

   9. fma-udef 99.5%

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

   10. neg-mul-1 99.5%

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

   11. +-commutative 99.5%

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

   12. unsub-neg 99.5%

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

5. Simplified 99.5%

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

6. Final simplification 99.5%

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

Alternative 2: 60.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;\tan x \leq 1:\\ \;\;\;\;{\left(\frac{1}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (tan x) -1.0)
   -1.0
   (if (<= (tan x) 1.0) (pow (/ 1.0 (hypot 1.0 (tan x))) 2.0) -1.0)))

C

double code(double x) {
	double tmp;
	if (tan(x) <= -1.0) {
		tmp = -1.0;
	} else if (tan(x) <= 1.0) {
		tmp = pow((1.0 / hypot(1.0, tan(x))), 2.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.tan(x) <= -1.0) {
		tmp = -1.0;
	} else if (Math.tan(x) <= 1.0) {
		tmp = Math.pow((1.0 / Math.hypot(1.0, Math.tan(x))), 2.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.tan(x) <= -1.0:
		tmp = -1.0
	elif math.tan(x) <= 1.0:
		tmp = math.pow((1.0 / math.hypot(1.0, math.tan(x))), 2.0)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (tan(x) <= -1.0)
		tmp = -1.0;
	elseif (tan(x) <= 1.0)
		tmp = Float64(1.0 / hypot(1.0, tan(x))) ^ 2.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (tan(x) <= -1.0)
		tmp = -1.0;
	elseif (tan(x) <= 1.0)
		tmp = (1.0 / hypot(1.0, tan(x))) ^ 2.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Tan[x], $MachinePrecision], -1.0], -1.0, If[LessEqual[N[Tan[x], $MachinePrecision], 1.0], N[Power[N[(1.0 / N[Sqrt[1.0 ^ 2 + N[Tan[x], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -1 or 1 < (tan.f64 x)

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-def 99.4%

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

   3. Simplified 99.4%

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

      1. add-log-exp 96.7%

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

      2. *-un-lft-identity 96.7%

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

      3. log-prod 96.7%

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

      4. metadata-eval 96.7%

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

      5. add-log-exp 99.4%

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

      6. pow2 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. +-lft-identity 99.4%

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

   7. Simplified 99.4%

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

   9. Applied egg-rr 20.6%

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

       1. *-inverses 20.6%

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

       2. metadata-eval 20.6%

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

       3. metadata-eval 20.6%

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

   11. Simplified 20.6%

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

   if -1 < (tan.f64 x) < 1

   1. Initial program 99.5%

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

      1. frac-2neg 99.5%

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

      2. div-inv 99.5%

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

      3. pow2 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-neg-in 99.5%

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

      6. neg-mul-1 99.5%

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

      7. metadata-eval 99.5%

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

      8. fma-def 99.5%

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

      9. pow2 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. neg-sub0 99.5%

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

      4. associate--r- 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

      7. unpow2 99.5%

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

      8. fma-udef 99.6%

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

      9. fma-udef 99.6%

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

      10. neg-mul-1 99.6%

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

      11. +-commutative 99.6%

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

      12. unsub-neg 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 69.7%

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

      1. sub-neg 69.7%

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

      2. metadata-eval 69.7%

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

      3. distribute-neg-in 69.7%

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

      4. metadata-eval 69.7%

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

      5. frac-2neg 69.7%

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

      6. add-sqr-sqrt 69.7%

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

      7. pow2 69.7%

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

      8. sqrt-div 69.7%

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

      9. metadata-eval 69.7%

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

      10. pow2 69.7%

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

      11. hypot-1-def 69.7%

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

   8. Applied egg-rr 69.7%

      \[\leadsto \color{blue}{{\left(\frac{1}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;\tan x \leq 1:\\ \;\;\;\;{\left(\frac{1}{\mathsf{hypot}\left(1, \tan x\right)}\right)}^{2}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 3: 99.5% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(1.0 - N[Power[N[Tan[x], $MachinePrecision], 2.0], $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. Step-by-step derivation

   1. add-log-exp 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. log-prod 98.7%

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

   4. metadata-eval 98.7%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

5. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

8. Final simplification 99.5%

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

Alternative 4: 60.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;\tan x \leq 1:\\ \;\;\;\;{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (tan x) -1.0)
   -1.0
   (if (<= (tan x) 1.0) (pow (hypot 1.0 (tan x)) -2.0) -1.0)))

C

double code(double x) {
	double tmp;
	if (tan(x) <= -1.0) {
		tmp = -1.0;
	} else if (tan(x) <= 1.0) {
		tmp = pow(hypot(1.0, tan(x)), -2.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.tan(x) <= -1.0) {
		tmp = -1.0;
	} else if (Math.tan(x) <= 1.0) {
		tmp = Math.pow(Math.hypot(1.0, Math.tan(x)), -2.0);
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.tan(x) <= -1.0:
		tmp = -1.0
	elif math.tan(x) <= 1.0:
		tmp = math.pow(math.hypot(1.0, math.tan(x)), -2.0)
	else:
		tmp = -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (tan(x) <= -1.0)
		tmp = -1.0;
	elseif (tan(x) <= 1.0)
		tmp = hypot(1.0, tan(x)) ^ -2.0;
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (tan(x) <= -1.0)
		tmp = -1.0;
	elseif (tan(x) <= 1.0)
		tmp = hypot(1.0, tan(x)) ^ -2.0;
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Tan[x], $MachinePrecision], -1.0], -1.0, If[LessEqual[N[Tan[x], $MachinePrecision], 1.0], N[Power[N[Sqrt[1.0 ^ 2 + N[Tan[x], $MachinePrecision] ^ 2], $MachinePrecision], -2.0], $MachinePrecision], -1.0]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -1 or 1 < (tan.f64 x)

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-def 99.4%

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

   3. Simplified 99.4%

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

      1. add-log-exp 96.7%

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

      2. *-un-lft-identity 96.7%

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

      3. log-prod 96.7%

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

      4. metadata-eval 96.7%

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

      5. add-log-exp 99.4%

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

      6. pow2 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. +-lft-identity 99.4%

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

   7. Simplified 99.4%

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

   9. Applied egg-rr 20.6%

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

       1. *-inverses 20.6%

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

       2. metadata-eval 20.6%

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

       3. metadata-eval 20.6%

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

   11. Simplified 20.6%

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

   if -1 < (tan.f64 x) < 1

   1. Initial program 99.5%

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

      1. frac-2neg 99.5%

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

      2. div-inv 99.5%

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

      3. pow2 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-neg-in 99.5%

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

      6. neg-mul-1 99.5%

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

      7. metadata-eval 99.5%

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

      8. fma-def 99.5%

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

      9. pow2 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. neg-sub0 99.5%

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

      4. associate--r- 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

      7. unpow2 99.5%

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

      8. fma-udef 99.6%

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

      9. fma-udef 99.6%

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

      10. neg-mul-1 99.6%

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

      11. +-commutative 99.6%

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

      12. unsub-neg 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 69.7%

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

      1. sub-neg 69.7%

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

      2. metadata-eval 69.7%

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

      3. distribute-neg-in 69.7%

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

      4. metadata-eval 69.7%

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

      5. frac-2neg 69.7%

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

      6. add-sqr-sqrt 69.7%

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

      7. sqrt-div 69.7%

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

      8. metadata-eval 69.7%

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

      9. pow2 69.7%

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

      10. hypot-1-def 69.7%

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

      11. sqrt-div 69.7%

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

      12. metadata-eval 69.7%

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

      13. pow2 69.7%

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

      14. hypot-1-def 69.7%

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

   8. Applied egg-rr 69.7%

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

      1. unpow-1 69.7%

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

      2. unpow-1 69.7%

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

      3. pow-sqr 69.7%

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

      4. metadata-eval 69.7%

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

   10. Simplified 69.7%

       \[\leadsto \color{blue}{{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;\tan x \leq 1:\\ \;\;\;\;{\left(\mathsf{hypot}\left(1, \tan x\right)\right)}^{-2}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 5: 60.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;\tan x \leq 1:\\ \;\;\;\;\frac{-1}{-1 - {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x) {
	double tmp;
	if (tan(x) <= -1.0) {
		tmp = -1.0;
	} else if (tan(x) <= 1.0) {
		tmp = -1.0 / (-1.0 - pow(tan(x), 2.0));
	} else {
		tmp = -1.0;
	}
	return tmp;
}

Fortran

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

Java

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

Python

def code(x):
	tmp = 0
	if math.tan(x) <= -1.0:
		tmp = -1.0
	elif math.tan(x) <= 1.0:
		tmp = -1.0 / (-1.0 - math.pow(math.tan(x), 2.0))
	else:
		tmp = -1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (tan(x) <= -1.0)
		tmp = -1.0;
	elseif (tan(x) <= 1.0)
		tmp = Float64(-1.0 / Float64(-1.0 - (tan(x) ^ 2.0)));
	else
		tmp = -1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (tan(x) <= -1.0)
		tmp = -1.0;
	elseif (tan(x) <= 1.0)
		tmp = -1.0 / (-1.0 - (tan(x) ^ 2.0));
	else
		tmp = -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (tan.f64 x) < -1 or 1 < (tan.f64 x)

   1. Initial program 99.4%

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

      1. +-commutative 99.4%

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

      2. fma-def 99.4%

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

   3. Simplified 99.4%

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

      1. add-log-exp 96.7%

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

      2. *-un-lft-identity 96.7%

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

      3. log-prod 96.7%

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

      4. metadata-eval 96.7%

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

      5. add-log-exp 99.4%

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

      6. pow2 99.4%

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

   5. Applied egg-rr 99.4%

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

      1. +-lft-identity 99.4%

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

   7. Simplified 99.4%

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

   9. Applied egg-rr 20.6%

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

       1. *-inverses 20.6%

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

       2. metadata-eval 20.6%

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

       3. metadata-eval 20.6%

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

   11. Simplified 20.6%

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

   if -1 < (tan.f64 x) < 1

   1. Initial program 99.5%

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

      1. frac-2neg 99.5%

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

      2. div-inv 99.5%

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

      3. pow2 99.5%

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

      4. +-commutative 99.5%

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

      5. distribute-neg-in 99.5%

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

      6. neg-mul-1 99.5%

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

      7. metadata-eval 99.5%

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

      8. fma-def 99.5%

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

      9. pow2 99.5%

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

   3. Applied egg-rr 99.5%

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

      1. associate-*r/ 99.5%

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

      2. *-rgt-identity 99.5%

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

      3. neg-sub0 99.5%

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

      4. associate--r- 99.5%

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

      5. metadata-eval 99.5%

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

      6. +-commutative 99.5%

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

      7. unpow2 99.5%

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

      8. fma-udef 99.6%

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

      9. fma-udef 99.6%

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

      10. neg-mul-1 99.6%

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

      11. +-commutative 99.6%

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

      12. unsub-neg 99.6%

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

   5. Simplified 99.6%

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

   6. Taylor expanded in x around 0 69.7%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan x \leq -1:\\ \;\;\;\;-1\\ \mathbf{elif}\;\tan x \leq 1:\\ \;\;\;\;\frac{-1}{-1 - {\tan x}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-1\\ \end{array} \]

Alternative 6: 99.5% 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. frac-2neg 99.5%

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

   2. div-inv 99.4%

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

   3. pow2 99.4%

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

   4. +-commutative 99.4%

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

   5. distribute-neg-in 99.4%

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

   6. neg-mul-1 99.4%

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

   7. metadata-eval 99.4%

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

   8. fma-def 99.4%

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

   9. pow2 99.4%

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

3. Applied egg-rr 99.4%

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

   1. associate-*r/ 99.5%

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

   2. *-rgt-identity 99.5%

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

   3. neg-sub0 99.5%

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

   4. associate--r- 99.5%

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

   5. metadata-eval 99.5%

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

   6. +-commutative 99.5%

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

   7. unpow2 99.5%

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

   8. fma-udef 99.5%

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

   9. fma-udef 99.5%

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

   10. neg-mul-1 99.5%

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

   11. +-commutative 99.5%

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

   12. unsub-neg 99.5%

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

5. Simplified 99.5%

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

   1. expm1-log1p-u 99.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - {\tan x}^{2}}\right)\right)} \]

   2. expm1-udef 99.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(\tan x, \tan x, -1\right)}{-1 - {\tan x}^{2}}\right)} - 1} \]

   3. fma-udef 99.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\tan x \cdot \tan x + -1}}{-1 - {\tan x}^{2}}\right)} - 1 \]

   4. pow2 99.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{{\tan x}^{2}} + -1}{-1 - {\tan x}^{2}}\right)} - 1 \]

7. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{\tan x}^{2} + -1}{-1 - {\tan x}^{2}}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 99.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{\tan x}^{2} + -1}{-1 - {\tan x}^{2}}\right)\right)} \]

   2. expm1-log1p 99.5%

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

   3. *-lft-identity 99.5%

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

   4. metadata-eval 99.5%

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

   5. times-frac 99.5%

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

   6. +-commutative 99.5%

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

   7. neg-mul-1 99.5%

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

   8. distribute-neg-in 99.5%

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

   9. metadata-eval 99.5%

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

   10. sub-neg 99.5%

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

   11. neg-mul-1 99.5%

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

   12. neg-sub0 99.5%

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

   13. associate--r- 99.5%

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

   14. metadata-eval 99.5%

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

9. Simplified 99.5%

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

10. Final simplification 99.5%

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

Alternative 7: 6.4% 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. +-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. Step-by-step derivation

   1. add-log-exp 98.7%

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

   2. *-un-lft-identity 98.7%

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

   3. log-prod 98.7%

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

   4. metadata-eval 98.7%

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

   5. add-log-exp 99.5%

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

   6. pow2 99.5%

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

5. Applied egg-rr 99.5%

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

   1. +-lft-identity 99.5%

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

7. Simplified 99.5%

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

9. Applied egg-rr 6.8%

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

    1. *-inverses 6.8%

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

    2. metadata-eval 6.8%

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

    3. metadata-eval 6.8%

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

11. Simplified 6.8%

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

12. Final simplification 6.8%

    \[\leadsto -1 \]

Alternative 8: 55.0% 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. Taylor expanded in x around 0 50.4%

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

3. Final simplification 50.4%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023189 
(FPCore (x)
  :name "Trigonometry B"
  :precision binary64
  (/ (- 1.0 (* (tan x) (tan x))) (+ 1.0 (* (tan x) (tan x)))))