tan-example (used to crash)

Percentage Accurate: 79.3% → 99.7%

Time: 26.7s

Alternatives: 12

Speedup: 1.0×

Specification

\[\left(\left(\left(x = 0 \lor 0.5884142 \leq x \land x \leq 505.5909\right) \land \left(-1.796658 \cdot 10^{+308} \leq y \land y \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq y \land y \leq 1.751224 \cdot 10^{+308}\right)\right) \land \left(-1.776707 \cdot 10^{+308} \leq z \land z \leq -8.599796 \cdot 10^{-310} \lor 3.293145 \cdot 10^{-311} \leq z \land z \leq 1.725154 \cdot 10^{+308}\right)\right) \land \left(-1.796658 \cdot 10^{+308} \leq a \land a \leq -9.425585 \cdot 10^{-310} \lor 1.284938 \cdot 10^{-309} \leq a \land a \leq 1.751224 \cdot 10^{+308}\right)\]

Math

\[\begin{array}{l} \\ x + \left(\tan \left(y + z\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (tan((y + z)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\tan \left(y + z\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (tan((y + z)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.7% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+
  x
  (-
   (/ (+ (tan y) (tan z)) (+ 1.0 (- -1.0 (fma (tan y) (tan z) -1.0))))
   (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) / (1.0 + (-1.0 - fma(tan(y), tan(z), -1.0)))) - tan(a));
}

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 + Float64(-1.0 - fma(tan(y), tan(z), -1.0)))) - tan(a)))
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(-1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. tan-sum 99.8%

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

   2. div-inv 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

5. Simplified 99.8%

   \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
6. Step-by-step derivation

   1. expm1-log1p-u 95.1%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan y \cdot \tan z\right)\right)}} - \tan a\right) \]

   2. expm1-udef 95.0%

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

   3. log1p-udef 95.0%

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

   4. add-exp-log 99.7%

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

7. Applied egg-rr 99.7%

   \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\left(\left(1 + \tan y \cdot \tan z\right) - 1\right)}} - \tan a\right) \]
8. Step-by-step derivation

   1. associate--l+ 99.8%

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

   2. fma-neg 99.8%

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

   3. metadata-eval 99.8%

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

9. Simplified 99.8%

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

10. Final simplification 99.8%

    \[\leadsto x + \left(\frac{\tan y + \tan z}{1 + \left(-1 - \mathsf{fma}\left(\tan y, \tan z, -1\right)\right)} - \tan a\right) \]

Alternative 2: 88.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02 \lor \neg \left(\tan a \leq 10^{-7}\right):\\ \;\;\;\;x + \left(\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}} - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan y + \tan z}{-\mathsf{fma}\left(\tan y, \tan z, -1\right)} + \left(x - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -0.02) (not (<= (tan a) 1e-7)))
   (+ x (- (/ 1.0 (/ (cos (+ y z)) (sin (+ y z)))) (tan a)))
   (+ (/ (+ (tan y) (tan z)) (- (fma (tan y) (tan z) -1.0))) (- x a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -0.02) || !(tan(a) <= 1e-7)) {
		tmp = x + ((1.0 / (cos((y + z)) / sin((y + z)))) - tan(a));
	} else {
		tmp = ((tan(y) + tan(z)) / -fma(tan(y), tan(z), -1.0)) + (x - a);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -0.02) || !(tan(a) <= 1e-7))
		tmp = Float64(x + Float64(Float64(1.0 / Float64(cos(Float64(y + z)) / sin(Float64(y + z)))) - tan(a)));
	else
		tmp = Float64(Float64(Float64(tan(y) + tan(z)) / Float64(-fma(tan(y), tan(z), -1.0))) + Float64(x - a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.02], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 1e-7]], $MachinePrecision]], N[(x + N[(N[(1.0 / N[(N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision] / N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / (-N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision] + -1.0), $MachinePrecision])), $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0200000000000000004 or 9.9999999999999995e-8 < (tan.f64 a)

   1. Initial program 81.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. tan-quot 81.3%

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      2. clear-num 81.3%

         \[\leadsto x + \left(\color{blue}{\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}}} - \tan a\right) \]

   3. Applied egg-rr 81.3%

      \[\leadsto x + \left(\color{blue}{\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}}} - \tan a\right) \]

   if -0.0200000000000000004 < (tan.f64 a) < 9.9999999999999995e-8

   1. Initial program 77.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. tan-sum 99.8%

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

      2. div-inv 99.8%

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

   3. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.8%

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

      2. *-rgt-identity 99.8%

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

   5. Simplified 99.8%

      \[\leadsto x + \left(\color{blue}{\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}} - \tan a\right) \]
   6. Step-by-step derivation

      1. expm1-log1p-u 93.5%

         \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan y \cdot \tan z\right)\right)}} - \tan a\right) \]

      2. expm1-udef 93.5%

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

      3. log1p-udef 93.5%

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

      4. add-exp-log 99.8%

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

   7. Applied egg-rr 99.8%

      \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \color{blue}{\left(\left(1 + \tan y \cdot \tan z\right) - 1\right)}} - \tan a\right) \]
   8. Step-by-step derivation

      1. associate--l+ 99.8%

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

      2. fma-neg 99.8%

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

      3. metadata-eval 99.8%

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

   9. Simplified 99.8%

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

       1. associate-+r- 99.8%

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

       2. associate--r+ 99.8%

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

       3. metadata-eval 99.8%

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

   11. Applied egg-rr 99.8%

       \[\leadsto \color{blue}{\left(x + \frac{\tan y + \tan z}{0 - \mathsf{fma}\left(\tan y, \tan z, -1\right)}\right) - \tan a} \]
   12. Step-by-step derivation

       1. +-commutative 99.8%

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

       2. associate--l+ 99.8%

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

       3. sub0-neg 99.8%

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

   13. Simplified 99.8%

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

   14. Taylor expanded in a around 0 99.8%

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

       1. +-commutative 99.8%

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

       2. mul-1-neg 99.8%

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

       3. unsub-neg 99.8%

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

   16. Simplified 99.8%

       \[\leadsto \frac{\tan y + \tan z}{-\mathsf{fma}\left(\tan y, \tan z, -1\right)} + \color{blue}{\left(x - a\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02 \lor \neg \left(\tan a \leq 10^{-7}\right):\\ \;\;\;\;x + \left(\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}} - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\tan y + \tan z}{-\mathsf{fma}\left(\tan y, \tan z, -1\right)} + \left(x - a\right)\\ \end{array} \]

Alternative 3: 88.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.06 \lor \neg \left(\tan a \leq 10^{-7}\right):\\ \;\;\;\;x + \left(\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}} - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -0.06) (not (<= (tan a) 1e-7)))
   (+ x (- (/ 1.0 (/ (cos (+ y z)) (sin (+ y z)))) (tan a)))
   (+ x (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -0.06) || !(tan(a) <= 1e-7)) {
		tmp = x + ((1.0 / (cos((y + z)) / sin((y + z)))) - tan(a));
	} else {
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((tan(a) <= (-0.06d0)) .or. (.not. (tan(a) <= 1d-7))) then
        tmp = x + ((1.0d0 / (cos((y + z)) / sin((y + z)))) - tan(a))
    else
        tmp = x + ((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((Math.tan(a) <= -0.06) || !(Math.tan(a) <= 1e-7)) {
		tmp = x + ((1.0 / (Math.cos((y + z)) / Math.sin((y + z)))) - Math.tan(a));
	} else {
		tmp = x + ((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z))));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -0.06) or not (math.tan(a) <= 1e-7):
		tmp = x + ((1.0 / (math.cos((y + z)) / math.sin((y + z)))) - math.tan(a))
	else:
		tmp = x + ((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z))))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -0.06) || !(tan(a) <= 1e-7))
		tmp = Float64(x + Float64(Float64(1.0 / Float64(cos(Float64(y + z)) / sin(Float64(y + z)))) - tan(a)));
	else
		tmp = Float64(x + Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((tan(a) <= -0.06) || ~((tan(a) <= 1e-7)))
		tmp = x + ((1.0 / (cos((y + z)) / sin((y + z)))) - tan(a));
	else
		tmp = x + ((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.06], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 1e-7]], $MachinePrecision]], N[(x + N[(N[(1.0 / N[(N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision] / N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.059999999999999998 or 9.9999999999999995e-8 < (tan.f64 a)

   1. Initial program 82.8%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. tan-quot 82.8%

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      2. clear-num 82.8%

         \[\leadsto x + \left(\color{blue}{\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}}} - \tan a\right) \]

   3. Applied egg-rr 82.8%

      \[\leadsto x + \left(\color{blue}{\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}}} - \tan a\right) \]

   if -0.059999999999999998 < (tan.f64 a) < 9.9999999999999995e-8

   1. Initial program 76.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. +-commutative 76.5%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) - \tan a\right) + x} \]

      2. associate-+l- 76.5%

         \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   3. Applied egg-rr 76.5%

      \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   4. Taylor expanded in a around 0 76.1%

      \[\leadsto \tan \left(y + z\right) - \color{blue}{-1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 76.1%

         \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]

   6. Simplified 76.1%

      \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]
   7. Step-by-step derivation

      1. sub-neg 76.1%

         \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(-\left(-x\right)\right)} \]

   8. Applied egg-rr 76.1%

      \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(-\left(-x\right)\right)} \]
   9. Step-by-step derivation

      1. remove-double-neg 76.1%

         \[\leadsto \tan \left(y + z\right) + \color{blue}{x} \]

      2. +-commutative 76.1%

         \[\leadsto \color{blue}{x + \tan \left(y + z\right)} \]

      3. +-commutative 76.1%

         \[\leadsto x + \tan \color{blue}{\left(z + y\right)} \]

   10. Simplified 76.1%

       \[\leadsto \color{blue}{x + \tan \left(z + y\right)} \]
   11. Step-by-step derivation

       1. tan-sum 97.0%

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

       2. +-commutative 97.0%

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

   12. Applied egg-rr 97.0%

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

4. Final simplification 90.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.06 \lor \neg \left(\tan a \leq 10^{-7}\right):\\ \;\;\;\;x + \left(\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}} - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{1 - \tan y \cdot \tan z}\\ \end{array} \]

Alternative 4: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - tan(a));
}

Fortran

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

Java

public static double code(double x, double y, double z, double a) {
	return x + (((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision] / N[(1.0 - N[(N[Tan[y], $MachinePrecision] * N[Tan[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. tan-sum 99.8%

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

   2. div-inv 99.8%

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

3. Applied egg-rr 99.8%

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

   1. associate-*r/ 99.8%

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

   2. *-rgt-identity 99.8%

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

5. Simplified 99.8%

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

6. Final simplification 99.8%

   \[\leadsto x + \left(\frac{\tan y + \tan z}{1 - \tan y \cdot \tan z} - \tan a\right) \]

Alternative 5: 60.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02 \lor \neg \left(\tan a \leq 10^{-12}\right):\\ \;\;\;\;x + \left(\sin z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -0.02) (not (<= (tan a) 1e-12)))
   (+ x (- (sin z) (tan a)))
   (+ (tan (+ y z)) (- x a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -0.02) || !(tan(a) <= 1e-12)) {
		tmp = x + (sin(z) - tan(a));
	} else {
		tmp = tan((y + z)) + (x - a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((tan(a) <= (-0.02d0)) .or. (.not. (tan(a) <= 1d-12))) then
        tmp = x + (sin(z) - tan(a))
    else
        tmp = tan((y + z)) + (x - a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((Math.tan(a) <= -0.02) || !(Math.tan(a) <= 1e-12)) {
		tmp = x + (Math.sin(z) - Math.tan(a));
	} else {
		tmp = Math.tan((y + z)) + (x - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -0.02) or not (math.tan(a) <= 1e-12):
		tmp = x + (math.sin(z) - math.tan(a))
	else:
		tmp = math.tan((y + z)) + (x - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -0.02) || !(tan(a) <= 1e-12))
		tmp = Float64(x + Float64(sin(z) - tan(a)));
	else
		tmp = Float64(tan(Float64(y + z)) + Float64(x - a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((tan(a) <= -0.02) || ~((tan(a) <= 1e-12)))
		tmp = x + (sin(z) - tan(a));
	else
		tmp = tan((y + z)) + (x - a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[Tan[a], $MachinePrecision], -0.02], N[Not[LessEqual[N[Tan[a], $MachinePrecision], 1e-12]], $MachinePrecision]], N[(x + N[(N[Sin[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] + N[(x - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0200000000000000004 or 9.9999999999999998e-13 < (tan.f64 a)

   1. Initial program 81.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. tan-quot 81.0%

         \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

      2. clear-num 81.0%

         \[\leadsto x + \left(\color{blue}{\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}}} - \tan a\right) \]

   3. Applied egg-rr 81.0%

      \[\leadsto x + \left(\color{blue}{\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}}} - \tan a\right) \]

   4. Taylor expanded in z around 0 64.0%

      \[\leadsto x + \left(\frac{1}{\frac{\color{blue}{-1 \cdot \left(z \cdot \sin y\right) + \cos y}}{\sin \left(y + z\right)}} - \tan a\right) \]
   5. Step-by-step derivation

      1. +-commutative 64.0%

         \[\leadsto x + \left(\frac{1}{\frac{\color{blue}{\cos y + -1 \cdot \left(z \cdot \sin y\right)}}{\sin \left(y + z\right)}} - \tan a\right) \]

      2. mul-1-neg 64.0%

         \[\leadsto x + \left(\frac{1}{\frac{\cos y + \color{blue}{\left(-z \cdot \sin y\right)}}{\sin \left(y + z\right)}} - \tan a\right) \]

      3. unsub-neg 64.0%

         \[\leadsto x + \left(\frac{1}{\frac{\color{blue}{\cos y - z \cdot \sin y}}{\sin \left(y + z\right)}} - \tan a\right) \]

   6. Simplified 64.0%

      \[\leadsto x + \left(\frac{1}{\frac{\color{blue}{\cos y - z \cdot \sin y}}{\sin \left(y + z\right)}} - \tan a\right) \]

   7. Taylor expanded in y around 0 46.2%

      \[\leadsto x + \left(\color{blue}{\sin z} - \tan a\right) \]

   if -0.0200000000000000004 < (tan.f64 a) < 9.9999999999999998e-13

   1. Initial program 78.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. +-commutative 78.1%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) - \tan a\right) + x} \]

      2. associate-+l- 78.1%

         \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   3. Applied egg-rr 78.1%

      \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   4. Taylor expanded in a around 0 78.1%

      \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(a + -1 \cdot x\right)} \]
   5. Step-by-step derivation

      1. neg-mul-1 78.1%

         \[\leadsto \tan \left(y + z\right) - \left(a + \color{blue}{\left(-x\right)}\right) \]

      2. unsub-neg 78.1%

         \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(a - x\right)} \]

   6. Simplified 78.1%

      \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(a - x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -0.02 \lor \neg \left(\tan a \leq 10^{-12}\right):\\ \;\;\;\;x + \left(\sin z - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(y + z\right) + \left(x - a\right)\\ \end{array} \]

Alternative 6: 79.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ x + \left(\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}} - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ 1.0 (/ (cos (+ y z)) (sin (+ y z)))) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + ((1.0 / (cos((y + z)) / sin((y + z)))) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + ((1.0d0 / (cos((y + z)) / sin((y + z)))) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + ((1.0 / (Math.cos((y + z)) / Math.sin((y + z)))) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + ((1.0 / (math.cos((y + z)) / math.sin((y + z)))) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(Float64(1.0 / Float64(cos(Float64(y + z)) / sin(Float64(y + z)))) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + ((1.0 / (cos((y + z)) / sin((y + z)))) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[(1.0 / N[(N[Cos[N[(y + z), $MachinePrecision]], $MachinePrecision] / N[Sin[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. tan-quot 79.5%

      \[\leadsto x + \left(\color{blue}{\frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}} - \tan a\right) \]

   2. clear-num 79.5%

      \[\leadsto x + \left(\color{blue}{\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}}} - \tan a\right) \]

3. Applied egg-rr 79.5%

   \[\leadsto x + \left(\color{blue}{\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}}} - \tan a\right) \]

4. Final simplification 79.5%

   \[\leadsto x + \left(\frac{1}{\frac{\cos \left(y + z\right)}{\sin \left(y + z\right)}} - \tan a\right) \]

Alternative 7: 60.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq 0.05:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) 0.05) (+ x (- (tan y) (tan a))) (+ x (tan (+ y z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 0.05) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + tan((y + z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((y + z) <= 0.05d0) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + tan((y + z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= 0.05) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + Math.tan((y + z));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= 0.05:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + math.tan((y + z))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= 0.05)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + tan(Float64(y + z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= 0.05)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + tan((y + z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], 0.05], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < 0.050000000000000003

   1. Initial program 85.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. +-commutative 85.3%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) - \tan a\right) + x} \]

      2. associate-+l- 85.2%

         \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   3. Applied egg-rr 85.2%

      \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   4. Taylor expanded in z around 0 68.2%

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} - \left(\tan a - x\right) \]
   5. Step-by-step derivation

      1. tan-quot 68.2%

         \[\leadsto \color{blue}{\tan y} - \left(\tan a - x\right) \]

      2. associate--r- 68.2%

         \[\leadsto \color{blue}{\left(\tan y - \tan a\right) + x} \]

   6. Applied egg-rr 68.2%

      \[\leadsto \color{blue}{\left(\tan y - \tan a\right) + x} \]

   if 0.050000000000000003 < (+.f64 y z)

   1. Initial program 69.3%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. +-commutative 69.3%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) - \tan a\right) + x} \]

      2. associate-+l- 69.3%

         \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   3. Applied egg-rr 69.3%

      \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   4. Taylor expanded in a around 0 44.4%

      \[\leadsto \tan \left(y + z\right) - \color{blue}{-1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 44.4%

         \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]

   6. Simplified 44.4%

      \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]
   7. Step-by-step derivation

      1. sub-neg 44.4%

         \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(-\left(-x\right)\right)} \]

   8. Applied egg-rr 44.4%

      \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(-\left(-x\right)\right)} \]
   9. Step-by-step derivation

      1. remove-double-neg 44.4%

         \[\leadsto \tan \left(y + z\right) + \color{blue}{x} \]

      2. +-commutative 44.4%

         \[\leadsto \color{blue}{x + \tan \left(y + z\right)} \]

      3. +-commutative 44.4%

         \[\leadsto x + \tan \color{blue}{\left(z + y\right)} \]

   10. Simplified 44.4%

       \[\leadsto \color{blue}{x + \tan \left(z + y\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq 0.05:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]

Alternative 8: 64.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.028:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sin z}{\cos z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 0.028) (+ x (- (tan y) (tan a))) (+ x (/ (sin z) (cos z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 0.028) {
		tmp = x + (tan(y) - tan(a));
	} else {
		tmp = x + (sin(z) / cos(z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= 0.028d0) then
        tmp = x + (tan(y) - tan(a))
    else
        tmp = x + (sin(z) / cos(z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 0.028) {
		tmp = x + (Math.tan(y) - Math.tan(a));
	} else {
		tmp = x + (Math.sin(z) / Math.cos(z));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if z <= 0.028:
		tmp = x + (math.tan(y) - math.tan(a))
	else:
		tmp = x + (math.sin(z) / math.cos(z))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (z <= 0.028)
		tmp = Float64(x + Float64(tan(y) - tan(a)));
	else
		tmp = Float64(x + Float64(sin(z) / cos(z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (z <= 0.028)
		tmp = x + (tan(y) - tan(a));
	else
		tmp = x + (sin(z) / cos(z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[z, 0.028], N[(x + N[(N[Tan[y], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Sin[z], $MachinePrecision] / N[Cos[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 0.0280000000000000006

   1. Initial program 88.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. +-commutative 88.1%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) - \tan a\right) + x} \]

      2. associate-+l- 88.0%

         \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   3. Applied egg-rr 88.0%

      \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   4. Taylor expanded in z around 0 74.1%

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} - \left(\tan a - x\right) \]
   5. Step-by-step derivation

      1. tan-quot 74.1%

         \[\leadsto \color{blue}{\tan y} - \left(\tan a - x\right) \]

      2. associate--r- 74.1%

         \[\leadsto \color{blue}{\left(\tan y - \tan a\right) + x} \]

   6. Applied egg-rr 74.1%

      \[\leadsto \color{blue}{\left(\tan y - \tan a\right) + x} \]

   if 0.0280000000000000006 < z

   1. Initial program 49.1%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. +-commutative 49.1%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) - \tan a\right) + x} \]

      2. associate-+l- 49.1%

         \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   3. Applied egg-rr 49.1%

      \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   4. Taylor expanded in a around 0 32.3%

      \[\leadsto \tan \left(y + z\right) - \color{blue}{-1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 32.3%

         \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]

   6. Simplified 32.3%

      \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]

   7. Taylor expanded in y around 0 32.1%

      \[\leadsto \color{blue}{\frac{\sin z}{\cos z} + x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.028:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\sin z}{\cos z}\\ \end{array} \]

Alternative 9: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(\tan \left(y + z\right) - \tan a\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (- (tan (+ y z)) (tan a))))

C

double code(double x, double y, double z, double a) {
	return x + (tan((y + z)) - tan(a));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + (tan((y + z)) - tan(a))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + (Math.tan((y + z)) - Math.tan(a));
}

Python

def code(x, y, z, a):
	return x + (math.tan((y + z)) - math.tan(a))

Julia

function code(x, y, z, a)
	return Float64(x + Float64(tan(Float64(y + z)) - tan(a)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + (tan((y + z)) - tan(a));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

2. Final simplification 79.5%

   \[\leadsto x + \left(\tan \left(y + z\right) - \tan a\right) \]

Alternative 10: 60.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -4 \cdot 10^{-5} \lor \neg \left(y + z \leq 0.05\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(\tan a - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (+ y z) -4e-5) (not (<= (+ y z) 0.05)))
   (+ x (tan (+ y z)))
   (- y (- (tan a) x))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (((y + z) <= -4e-5) || !((y + z) <= 0.05)) {
		tmp = x + tan((y + z));
	} else {
		tmp = y - (tan(a) - x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    real(8) :: tmp
    if (((y + z) <= (-4d-5)) .or. (.not. ((y + z) <= 0.05d0))) then
        tmp = x + tan((y + z))
    else
        tmp = y - (tan(a) - x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if (((y + z) <= -4e-5) || !((y + z) <= 0.05)) {
		tmp = x + Math.tan((y + z));
	} else {
		tmp = y - (Math.tan(a) - x);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if ((y + z) <= -4e-5) or not ((y + z) <= 0.05):
		tmp = x + math.tan((y + z))
	else:
		tmp = y - (math.tan(a) - x)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((Float64(y + z) <= -4e-5) || !(Float64(y + z) <= 0.05))
		tmp = Float64(x + tan(Float64(y + z)));
	else
		tmp = Float64(y - Float64(tan(a) - x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if (((y + z) <= -4e-5) || ~(((y + z) <= 0.05)))
		tmp = x + tan((y + z));
	else
		tmp = y - (tan(a) - x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[Or[LessEqual[N[(y + z), $MachinePrecision], -4e-5], N[Not[LessEqual[N[(y + z), $MachinePrecision], 0.05]], $MachinePrecision]], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(y - N[(N[Tan[a], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -4.00000000000000033e-5 or 0.050000000000000003 < (+.f64 y z)

   1. Initial program 72.0%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. +-commutative 72.0%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) - \tan a\right) + x} \]

      2. associate-+l- 72.0%

         \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   3. Applied egg-rr 72.0%

      \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   4. Taylor expanded in a around 0 47.0%

      \[\leadsto \tan \left(y + z\right) - \color{blue}{-1 \cdot x} \]
   5. Step-by-step derivation

      1. neg-mul-1 47.0%

         \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]

   6. Simplified 47.0%

      \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]
   7. Step-by-step derivation

      1. sub-neg 47.0%

         \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(-\left(-x\right)\right)} \]

   8. Applied egg-rr 47.0%

      \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(-\left(-x\right)\right)} \]
   9. Step-by-step derivation

      1. remove-double-neg 47.0%

         \[\leadsto \tan \left(y + z\right) + \color{blue}{x} \]

      2. +-commutative 47.0%

         \[\leadsto \color{blue}{x + \tan \left(y + z\right)} \]

      3. +-commutative 47.0%

         \[\leadsto x + \tan \color{blue}{\left(z + y\right)} \]

   10. Simplified 47.0%

       \[\leadsto \color{blue}{x + \tan \left(z + y\right)} \]

   if -4.00000000000000033e-5 < (+.f64 y z) < 0.050000000000000003

   1. Initial program 99.9%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Step-by-step derivation

      1. +-commutative 99.9%

         \[\leadsto \color{blue}{\left(\tan \left(y + z\right) - \tan a\right) + x} \]

      2. associate-+l- 99.9%

         \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

   4. Taylor expanded in z around 0 98.9%

      \[\leadsto \color{blue}{\frac{\sin y}{\cos y}} - \left(\tan a - x\right) \]

   5. Taylor expanded in y around 0 98.9%

      \[\leadsto \color{blue}{y} - \left(\tan a - x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -4 \cdot 10^{-5} \lor \neg \left(y + z \leq 0.05\right):\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;y - \left(\tan a - x\right)\\ \end{array} \]

Alternative 11: 50.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x + \tan \left(y + z\right) \end{array} \]

FPCore

(FPCore (x y z a) :precision binary64 (+ x (tan (+ y z))))

C

double code(double x, double y, double z, double a) {
	return x + tan((y + z));
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x + tan((y + z))
end function

Java

public static double code(double x, double y, double z, double a) {
	return x + Math.tan((y + z));
}

Python

def code(x, y, z, a):
	return x + math.tan((y + z))

Julia

function code(x, y, z, a)
	return Float64(x + tan(Float64(y + z)))
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x + tan((y + z));
end

Wolfram

code[x_, y_, z_, a_] := N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 79.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Step-by-step derivation

   1. +-commutative 79.5%

      \[\leadsto \color{blue}{\left(\tan \left(y + z\right) - \tan a\right) + x} \]

   2. associate-+l- 79.5%

      \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

3. Applied egg-rr 79.5%

   \[\leadsto \color{blue}{\tan \left(y + z\right) - \left(\tan a - x\right)} \]

4. Taylor expanded in a around 0 50.3%

   \[\leadsto \tan \left(y + z\right) - \color{blue}{-1 \cdot x} \]
5. Step-by-step derivation

   1. neg-mul-1 50.3%

      \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]

6. Simplified 50.3%

   \[\leadsto \tan \left(y + z\right) - \color{blue}{\left(-x\right)} \]
7. Step-by-step derivation

   1. sub-neg 50.3%

      \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(-\left(-x\right)\right)} \]

8. Applied egg-rr 50.3%

   \[\leadsto \color{blue}{\tan \left(y + z\right) + \left(-\left(-x\right)\right)} \]
9. Step-by-step derivation

   1. remove-double-neg 50.3%

      \[\leadsto \tan \left(y + z\right) + \color{blue}{x} \]

   2. +-commutative 50.3%

      \[\leadsto \color{blue}{x + \tan \left(y + z\right)} \]

   3. +-commutative 50.3%

      \[\leadsto x + \tan \color{blue}{\left(z + y\right)} \]

10. Simplified 50.3%

    \[\leadsto \color{blue}{x + \tan \left(z + y\right)} \]

11. Final simplification 50.3%

    \[\leadsto x + \tan \left(y + z\right) \]

Alternative 12: 32.0% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x y z a) :precision binary64 x)

C

double code(double x, double y, double z, double a) {
	return x;
}

Fortran

real(8) function code(x, y, z, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: a
    code = x
end function

Java

public static double code(double x, double y, double z, double a) {
	return x;
}

Python

def code(x, y, z, a):
	return x

Julia

function code(x, y, z, a)
	return x
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = x;
end

Wolfram

code[x_, y_, z_, a_] := x

Derivation

1. Initial program 79.5%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]

2. Taylor expanded in x around inf 32.2%

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

3. Final simplification 32.2%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023192 
(FPCore (x y z a)
  :name "tan-example (used to crash)"
  :precision binary64
  :pre (and (and (and (or (== x 0.0) (and (<= 0.5884142 x) (<= x 505.5909))) (or (and (<= -1.796658e+308 y) (<= y -9.425585e-310)) (and (<= 1.284938e-309 y) (<= y 1.751224e+308)))) (or (and (<= -1.776707e+308 z) (<= z -8.599796e-310)) (and (<= 3.293145e-311 z) (<= z 1.725154e+308)))) (or (and (<= -1.796658e+308 a) (<= a -9.425585e-310)) (and (<= 1.284938e-309 a) (<= a 1.751224e+308))))
  (+ x (- (tan (+ y z)) (tan a))))