tan-example (used to crash)

Percentage Accurate: 79.3% → 99.7%

Time: 37.8s

Alternatives: 10

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 75.0%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. +-commutative 99.8%

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

   2. tan-quot 99.8%

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

   3. div-inv 99.8%

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

   4. fma-def 99.7%

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

7. Applied egg-rr 99.7%

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

   1. fma-udef 99.8%

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

   2. associate-*r/ 99.8%

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

   3. *-rgt-identity 99.8%

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

9. Simplified 99.8%

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

    1. div-inv 99.8%

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

    2. fma-neg 99.8%

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

    3. tan-quot 99.8%

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

    4. +-commutative 99.8%

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

11. Applied egg-rr 99.8%

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

    1. sub-neg 99.8%

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

    2. +-commutative 99.8%

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

    3. distribute-rgt-neg-in 99.8%

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

    4. fma-def 99.8%

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

13. Simplified 99.8%

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

14. Final simplification 99.8%

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

Alternative 2: 89.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -0.02) (not (<= (tan a) 1e-10)))
   (+ x (- (+ (tan y) (/ (sin z) (cos z))) (tan a)))
   (+ x (- (/ (+ (tan y) (tan z)) (fma (tan y) (- (tan z)) 1.0)) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -0.02) || !(tan(a) <= 1e-10)) {
		tmp = x + ((tan(y) + (sin(z) / cos(z))) - tan(a));
	} else {
		tmp = x + (((tan(y) + tan(z)) / fma(tan(y), -tan(z), 1.0)) - a);
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -0.02) || !(tan(a) <= 1e-10))
		tmp = Float64(x + Float64(Float64(tan(y) + Float64(sin(z) / cos(z))) - tan(a)));
	else
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / fma(tan(y), Float64(-tan(z)), 1.0)) - 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-10]], $MachinePrecision]], N[(x + N[(N[(N[Tan[y], $MachinePrecision] + N[(N[Sin[z], $MachinePrecision] / N[Cos[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + 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] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0200000000000000004 or 1.00000000000000004e-10 < (tan.f64 a)

   1. Initial program 75.1%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   5. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. tan-quot 99.7%

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

      3. div-inv 99.7%

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

      4. fma-def 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. fma-udef 99.7%

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

      2. associate-*r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in y around 0 75.7%

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

   if -0.0200000000000000004 < (tan.f64 a) < 1.00000000000000004e-10

   1. Initial program 74.9%

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

   2. Taylor expanded in a around 0 74.9%

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

      1. tan-sum 99.9%

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

      2. div-inv 99.9%

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

      3. fma-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. unsub-neg 99.9%

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

      3. associate-*r/ 99.9%

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

      4. *-rgt-identity 99.9%

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

   6. Simplified 99.9%

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

      1. sub-neg 99.9%

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

   8. Applied egg-rr 99.9%

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

      1. +-commutative 99.9%

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

      2. distribute-rgt-neg-in 99.9%

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

      3. fma-def 99.9%

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

   10. Simplified 99.9%

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

4. Final simplification 86.8%

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

Alternative 3: 89.6% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (or (<= (tan a) -0.02) (not (<= (tan a) 1e-10)))
   (+ x (- (+ (tan y) (/ (sin z) (cos z))) (tan a)))
   (+ x (- (/ (+ (tan y) (tan z)) (- 1.0 (* (tan y) (tan z)))) a))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((tan(a) <= -0.02) || !(tan(a) <= 1e-10)) {
		tmp = x + ((tan(y) + (sin(z) / cos(z))) - tan(a));
	} else {
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - 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-10))) then
        tmp = x + ((tan(y) + (sin(z) / cos(z))) - tan(a))
    else
        tmp = x + (((tan(y) + tan(z)) / (1.0d0 - (tan(y) * tan(z)))) - 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-10)) {
		tmp = x + ((Math.tan(y) + (Math.sin(z) / Math.cos(z))) - Math.tan(a));
	} else {
		tmp = x + (((Math.tan(y) + Math.tan(z)) / (1.0 - (Math.tan(y) * Math.tan(z)))) - a);
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (math.tan(a) <= -0.02) or not (math.tan(a) <= 1e-10):
		tmp = x + ((math.tan(y) + (math.sin(z) / math.cos(z))) - math.tan(a))
	else:
		tmp = x + (((math.tan(y) + math.tan(z)) / (1.0 - (math.tan(y) * math.tan(z)))) - a)
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if ((tan(a) <= -0.02) || !(tan(a) <= 1e-10))
		tmp = Float64(x + Float64(Float64(tan(y) + Float64(sin(z) / cos(z))) - tan(a)));
	else
		tmp = Float64(x + Float64(Float64(Float64(tan(y) + tan(z)) / Float64(1.0 - Float64(tan(y) * tan(z)))) - 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-10)))
		tmp = x + ((tan(y) + (sin(z) / cos(z))) - tan(a));
	else
		tmp = x + (((tan(y) + tan(z)) / (1.0 - (tan(y) * tan(z)))) - 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-10]], $MachinePrecision]], N[(x + N[(N[(N[Tan[y], $MachinePrecision] + N[(N[Sin[z], $MachinePrecision] / N[Cos[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 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] - a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (tan.f64 a) < -0.0200000000000000004 or 1.00000000000000004e-10 < (tan.f64 a)

   1. Initial program 75.1%

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

      1. tan-sum 99.7%

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

      2. div-inv 99.7%

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

   3. Applied egg-rr 99.7%

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

      1. associate-*r/ 99.7%

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

      2. *-rgt-identity 99.7%

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

   5. Simplified 99.7%

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

      1. +-commutative 99.7%

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

      2. tan-quot 99.7%

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

      3. div-inv 99.7%

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

      4. fma-def 99.7%

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

   7. Applied egg-rr 99.7%

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

      1. fma-udef 99.7%

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

      2. associate-*r/ 99.7%

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

      3. *-rgt-identity 99.7%

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

   9. Simplified 99.7%

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

   10. Taylor expanded in y around 0 75.7%

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

   if -0.0200000000000000004 < (tan.f64 a) < 1.00000000000000004e-10

   1. Initial program 74.9%

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

   2. Taylor expanded in a around 0 74.9%

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

      1. tan-sum 99.9%

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

      2. div-inv 99.9%

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

      3. fma-neg 99.9%

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

   4. Applied egg-rr 99.9%

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

      1. fma-udef 99.9%

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

      2. unsub-neg 99.9%

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

      3. associate-*r/ 99.9%

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

      4. *-rgt-identity 99.9%

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

   6. Simplified 99.9%

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

4. Final simplification 86.8%

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

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 75.0%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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: 79.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 75.0%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
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. +-commutative 99.8%

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

   2. tan-quot 99.8%

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

   3. div-inv 99.8%

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

   4. fma-def 99.7%

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

7. Applied egg-rr 99.7%

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

   1. fma-udef 99.8%

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

   2. associate-*r/ 99.8%

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

   3. *-rgt-identity 99.8%

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

9. Simplified 99.8%

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

10. Taylor expanded in y around 0 75.6%

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

11. Final simplification 75.6%

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

Alternative 6: 60.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -1 \cdot 10^{-5}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z + \left(x - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -1e-5) (+ x (tan (+ y z))) (+ (tan z) (- x (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -1e-5) {
		tmp = x + tan((y + z));
	} else {
		tmp = tan(z) + (x - tan(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 ((y + z) <= (-1d-5)) then
        tmp = x + tan((y + z))
    else
        tmp = tan(z) + (x - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -1e-5) {
		tmp = x + Math.tan((y + z));
	} else {
		tmp = Math.tan(z) + (x - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -1e-5:
		tmp = x + math.tan((y + z))
	else:
		tmp = math.tan(z) + (x - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -1e-5)
		tmp = Float64(x + tan(Float64(y + z)));
	else
		tmp = Float64(tan(z) + Float64(x - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -1e-5)
		tmp = x + tan((y + z));
	else
		tmp = tan(z) + (x - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -1e-5], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Tan[z], $MachinePrecision] + N[(x - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -1.00000000000000008e-5

   1. Initial program 71.0%

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

      1. associate-+r- 70.8%

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

      2. +-commutative 70.8%

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

      3. associate--l+ 70.8%

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

   3. Simplified 70.8%

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

      1. add-cube-cbrt 69.9%

         \[\leadsto \tan \left(y + z\right) + \color{blue}{\left(\sqrt[3]{x - \tan a} \cdot \sqrt[3]{x - \tan a}\right) \cdot \sqrt[3]{x - \tan a}} \]

      2. pow3 69.9%

         \[\leadsto \tan \left(y + z\right) + \color{blue}{{\left(\sqrt[3]{x - \tan a}\right)}^{3}} \]

   5. Applied egg-rr 69.9%

      \[\leadsto \tan \left(y + z\right) + \color{blue}{{\left(\sqrt[3]{x - \tan a}\right)}^{3}} \]

   6. Taylor expanded in a around 0 41.0%

      \[\leadsto \tan \left(y + z\right) + {\color{blue}{\left({x}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 40.8%

         \[\leadsto \tan \left(y + z\right) + {\color{blue}{\left(\sqrt[3]{x}\right)}}^{3} \]

   8. Simplified 40.8%

      \[\leadsto \tan \left(y + z\right) + {\color{blue}{\left(\sqrt[3]{x}\right)}}^{3} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 40.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \left(y + z\right) + {\left(\sqrt[3]{x}\right)}^{3}\right)\right)} \]

      2. expm1-udef 40.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \left(y + z\right) + {\left(\sqrt[3]{x}\right)}^{3}\right)} - 1} \]

      3. +-commutative 40.3%

         \[\leadsto e^{\mathsf{log1p}\left(\tan \color{blue}{\left(z + y\right)} + {\left(\sqrt[3]{x}\right)}^{3}\right)} - 1 \]

      4. rem-cube-cbrt 40.6%

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

   10. Applied egg-rr 40.6%

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

       1. expm1-def 40.5%

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

       2. expm1-log1p 41.4%

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

       3. +-commutative 41.4%

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

       4. +-commutative 41.4%

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

   12. Simplified 41.4%

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

   if -1.00000000000000008e-5 < (+.f64 y z)

   1. Initial program 77.9%

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

      1. associate-+r- 77.9%

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

      2. +-commutative 77.9%

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

      3. associate--l+ 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in y around 0 60.7%

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

      1. tan-quot 60.7%

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

      2. expm1-log1p-u 54.5%

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

      3. expm1-udef 54.6%

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

   6. Applied egg-rr 54.6%

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

      1. expm1-def 54.5%

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

      2. expm1-log1p 60.7%

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

   8. Simplified 60.7%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -1 \cdot 10^{-5}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;\tan z + \left(x - \tan a\right)\\ \end{array} \]

Alternative 7: 60.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y + z \leq -1 \cdot 10^{-5}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= (+ y z) -1e-5) (+ x (tan (+ y z))) (+ x (- (tan z) (tan a)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -1e-5) {
		tmp = x + tan((y + z));
	} else {
		tmp = x + (tan(z) - tan(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 ((y + z) <= (-1d-5)) then
        tmp = x + tan((y + z))
    else
        tmp = x + (tan(z) - tan(a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double a) {
	double tmp;
	if ((y + z) <= -1e-5) {
		tmp = x + Math.tan((y + z));
	} else {
		tmp = x + (Math.tan(z) - Math.tan(a));
	}
	return tmp;
}

Python

def code(x, y, z, a):
	tmp = 0
	if (y + z) <= -1e-5:
		tmp = x + math.tan((y + z))
	else:
		tmp = x + (math.tan(z) - math.tan(a))
	return tmp

Julia

function code(x, y, z, a)
	tmp = 0.0
	if (Float64(y + z) <= -1e-5)
		tmp = Float64(x + tan(Float64(y + z)));
	else
		tmp = Float64(x + Float64(tan(z) - tan(a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, a)
	tmp = 0.0;
	if ((y + z) <= -1e-5)
		tmp = x + tan((y + z));
	else
		tmp = x + (tan(z) - tan(a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, a_] := If[LessEqual[N[(y + z), $MachinePrecision], -1e-5], N[(x + N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[Tan[z], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 y z) < -1.00000000000000008e-5

   1. Initial program 71.0%

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

      1. associate-+r- 70.8%

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

      2. +-commutative 70.8%

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

      3. associate--l+ 70.8%

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

   3. Simplified 70.8%

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

      1. add-cube-cbrt 69.9%

         \[\leadsto \tan \left(y + z\right) + \color{blue}{\left(\sqrt[3]{x - \tan a} \cdot \sqrt[3]{x - \tan a}\right) \cdot \sqrt[3]{x - \tan a}} \]

      2. pow3 69.9%

         \[\leadsto \tan \left(y + z\right) + \color{blue}{{\left(\sqrt[3]{x - \tan a}\right)}^{3}} \]

   5. Applied egg-rr 69.9%

      \[\leadsto \tan \left(y + z\right) + \color{blue}{{\left(\sqrt[3]{x - \tan a}\right)}^{3}} \]

   6. Taylor expanded in a around 0 41.0%

      \[\leadsto \tan \left(y + z\right) + {\color{blue}{\left({x}^{0.3333333333333333}\right)}}^{3} \]
   7. Step-by-step derivation

      1. unpow1/3 40.8%

         \[\leadsto \tan \left(y + z\right) + {\color{blue}{\left(\sqrt[3]{x}\right)}}^{3} \]

   8. Simplified 40.8%

      \[\leadsto \tan \left(y + z\right) + {\color{blue}{\left(\sqrt[3]{x}\right)}}^{3} \]
   9. Step-by-step derivation

      1. expm1-log1p-u 40.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \left(y + z\right) + {\left(\sqrt[3]{x}\right)}^{3}\right)\right)} \]

      2. expm1-udef 40.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \left(y + z\right) + {\left(\sqrt[3]{x}\right)}^{3}\right)} - 1} \]

      3. +-commutative 40.3%

         \[\leadsto e^{\mathsf{log1p}\left(\tan \color{blue}{\left(z + y\right)} + {\left(\sqrt[3]{x}\right)}^{3}\right)} - 1 \]

      4. rem-cube-cbrt 40.6%

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

   10. Applied egg-rr 40.6%

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

       1. expm1-def 40.5%

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

       2. expm1-log1p 41.4%

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

       3. +-commutative 41.4%

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

       4. +-commutative 41.4%

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

   12. Simplified 41.4%

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

   if -1.00000000000000008e-5 < (+.f64 y z)

   1. Initial program 77.9%

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

      1. associate-+r- 77.9%

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

      2. +-commutative 77.9%

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

      3. associate--l+ 77.9%

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

   3. Simplified 77.9%

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

   4. Taylor expanded in y around 0 60.7%

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

      1. add-exp-log 55.9%

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

      2. tan-quot 55.9%

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

   6. Applied egg-rr 55.9%

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

      1. add-exp-log 60.7%

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

      2. +-commutative 60.7%

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

      3. associate-+l- 60.7%

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

   8. Applied egg-rr 60.7%

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

4. Final simplification 52.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y + z \leq -1 \cdot 10^{-5}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(\tan z - \tan a\right)\\ \end{array} \]

Alternative 8: 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 75.0%

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

2. Final simplification 75.0%

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

Alternative 9: 50.2% 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 75.0%

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

   1. associate-+r- 74.9%

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

   2. +-commutative 74.9%

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

   3. associate--l+ 74.9%

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

3. Simplified 74.9%

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

   1. add-cube-cbrt 73.8%

      \[\leadsto \tan \left(y + z\right) + \color{blue}{\left(\sqrt[3]{x - \tan a} \cdot \sqrt[3]{x - \tan a}\right) \cdot \sqrt[3]{x - \tan a}} \]

   2. pow3 73.8%

      \[\leadsto \tan \left(y + z\right) + \color{blue}{{\left(\sqrt[3]{x - \tan a}\right)}^{3}} \]

5. Applied egg-rr 73.8%

   \[\leadsto \tan \left(y + z\right) + \color{blue}{{\left(\sqrt[3]{x - \tan a}\right)}^{3}} \]

6. Taylor expanded in a around 0 46.1%

   \[\leadsto \tan \left(y + z\right) + {\color{blue}{\left({x}^{0.3333333333333333}\right)}}^{3} \]
7. Step-by-step derivation

   1. unpow1/3 46.0%

      \[\leadsto \tan \left(y + z\right) + {\color{blue}{\left(\sqrt[3]{x}\right)}}^{3} \]

8. Simplified 46.0%

   \[\leadsto \tan \left(y + z\right) + {\color{blue}{\left(\sqrt[3]{x}\right)}}^{3} \]
9. Step-by-step derivation

   1. expm1-log1p-u 45.5%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\tan \left(y + z\right) + {\left(\sqrt[3]{x}\right)}^{3}\right)\right)} \]

   2. expm1-udef 45.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\tan \left(y + z\right) + {\left(\sqrt[3]{x}\right)}^{3}\right)} - 1} \]

   3. +-commutative 45.5%

      \[\leadsto e^{\mathsf{log1p}\left(\tan \color{blue}{\left(z + y\right)} + {\left(\sqrt[3]{x}\right)}^{3}\right)} - 1 \]

   4. rem-cube-cbrt 45.8%

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

10. Applied egg-rr 45.8%

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

    1. expm1-def 45.8%

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

    2. expm1-log1p 46.6%

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

    3. +-commutative 46.6%

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

    4. +-commutative 46.6%

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

12. Simplified 46.6%

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

13. Final simplification 46.6%

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

Alternative 10: 31.6% 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 75.0%

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

2. Taylor expanded in x around inf 28.8%

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

3. Final simplification 28.8%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023285 
(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))))