tan-example (used to crash)

Percentage Accurate: 79.2% → 99.7%

Time: 29.5s

Alternatives: 9

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: 79.2% 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

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

FPCore

(FPCore (x y z a)
 :precision binary64
 (+ x (- (/ (+ (tan y) (tan z)) (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)) / 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)) / fma(tan(y), Float64(-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[(N[Tan[y], $MachinePrecision] * (-N[Tan[z], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 80.9%

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

   1. tan-sum 99.6%

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

   2. div-inv 99.6%

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

4. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

6. Simplified 99.6%

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

   1. expm1-log1p-u 94.1%

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

   2. expm1-undefine 94.2%

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

   3. log1p-undefine 94.2%

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

   4. add-exp-log 99.6%

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

8. Applied egg-rr 99.6%

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

   1. associate--l+ 99.6%

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

   2. fma-neg 99.6%

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

   3. metadata-eval 99.6%

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

10. Simplified 99.6%

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

    1. sub-neg 99.6%

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

    2. associate--r+ 99.7%

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

    3. metadata-eval 99.7%

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

12. Applied egg-rr 99.7%

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

    1. *-rgt-identity 99.7%

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

    2. associate-*r/ 99.6%

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

    3. fma-undefine 99.7%

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

    4. sub0-neg 99.7%

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

    5. fma-undefine 99.6%

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

    6. distribute-neg-in 99.6%

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

    7. metadata-eval 99.6%

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

    8. +-commutative 99.6%

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

    9. sub-neg 99.6%

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

    10. fma-neg 99.6%

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

14. Simplified 99.7%

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

Alternative 2: 89.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \tan y + \tan z\\ \mathbf{if}\;\tan a \leq -8 \cdot 10^{-16}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{t\_0}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(t\_0 - \tan a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (let* ((t_0 (+ (tan y) (tan z))))
   (if (<= (tan a) -8e-16)
     (+ x (- (tan (+ y z)) (tan a)))
     (if (<= (tan a) 2e-11)
       (+ x (/ t_0 (fma (tan y) (- (tan z)) 1.0)))
       (+ x (- t_0 (tan a)))))))

C

double code(double x, double y, double z, double a) {
	double t_0 = tan(y) + tan(z);
	double tmp;
	if (tan(a) <= -8e-16) {
		tmp = x + (tan((y + z)) - tan(a));
	} else if (tan(a) <= 2e-11) {
		tmp = x + (t_0 / fma(tan(y), -tan(z), 1.0));
	} else {
		tmp = x + (t_0 - tan(a));
	}
	return tmp;
}

Julia

function code(x, y, z, a)
	t_0 = Float64(tan(y) + tan(z))
	tmp = 0.0
	if (tan(a) <= -8e-16)
		tmp = Float64(x + Float64(tan(Float64(y + z)) - tan(a)));
	elseif (tan(a) <= 2e-11)
		tmp = Float64(x + Float64(t_0 / fma(tan(y), Float64(-tan(z)), 1.0)));
	else
		tmp = Float64(x + Float64(t_0 - tan(a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, a_] := Block[{t$95$0 = N[(N[Tan[y], $MachinePrecision] + N[Tan[z], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Tan[a], $MachinePrecision], -8e-16], N[(x + N[(N[Tan[N[(y + z), $MachinePrecision]], $MachinePrecision] - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Tan[a], $MachinePrecision], 2e-11], N[(x + N[(t$95$0 / N[(N[Tan[y], $MachinePrecision] * (-N[Tan[z], $MachinePrecision]) + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(t$95$0 - N[Tan[a], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (tan.f64 a) < -7.9999999999999998e-16

   1. Initial program 80.5%

      \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
   2. Add Preprocessing

   if -7.9999999999999998e-16 < (tan.f64 a) < 1.99999999999999988e-11

   1. Initial program 80.9%

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

      1. expm1-log1p-u 76.8%

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

      2. +-commutative 76.8%

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

      3. associate-+l- 76.8%

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

   4. Applied egg-rr 76.8%

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

   5. Taylor expanded in a around 0 76.7%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. log1p-define 76.7%

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

   7. Simplified 76.7%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}\right) \]
   8. Step-by-step derivation

      1. expm1-log1p-u 80.8%

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

      2. tan-quot 80.9%

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

      3. +-commutative 80.9%

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

      4. tan-sum 99.5%

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

      5. div-inv 99.5%

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

      6. fma-define 99.5%

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

   9. Applied egg-rr 99.5%

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

       1. fma-undefine 99.5%

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

       2. +-commutative 99.5%

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

       3. associate-*r/ 99.5%

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

       4. *-rgt-identity 99.5%

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

       5. sub-neg 99.5%

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

       6. +-commutative 99.5%

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

       7. metadata-eval 99.5%

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

       8. sub-neg 99.5%

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

       9. distribute-rgt-neg-in 99.5%

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

       10. fma-neg 99.5%

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

       11. metadata-eval 99.5%

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

   11. Simplified 99.5%

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

   if 1.99999999999999988e-11 < (tan.f64 a)

   1. Initial program 81.1%

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

      1. tan-sum 99.3%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. associate-*r/ 99.3%

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

      2. *-rgt-identity 99.3%

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

   6. Simplified 99.3%

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

      1. expm1-log1p-u 95.2%

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

      2. expm1-undefine 95.3%

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

      3. log1p-undefine 95.3%

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

      4. add-exp-log 99.4%

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

   8. Applied egg-rr 99.4%

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

      1. associate--l+ 99.3%

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

      2. fma-neg 99.3%

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

      3. metadata-eval 99.3%

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

   10. Simplified 99.3%

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

       1. sub-neg 99.3%

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

       2. associate--r+ 99.3%

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

       3. metadata-eval 99.3%

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

   12. Applied egg-rr 99.3%

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

       1. *-rgt-identity 99.3%

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

       2. associate-*r/ 99.3%

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

       3. fma-undefine 99.3%

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

       4. sub0-neg 99.3%

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

       5. fma-undefine 99.3%

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

       6. distribute-neg-in 99.3%

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

       7. metadata-eval 99.3%

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

       8. +-commutative 99.3%

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

       9. sub-neg 99.3%

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

       10. fma-neg 99.3%

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

   14. Simplified 99.3%

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

   15. Taylor expanded in y around 0 82.0%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\tan a \leq -8 \cdot 10^{-16}:\\ \;\;\;\;x + \left(\tan \left(y + z\right) - \tan a\right)\\ \mathbf{elif}\;\tan a \leq 2 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{\tan y + \tan z}{\mathsf{fma}\left(\tan y, -\tan z, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;x + \left(\left(\tan y + \tan z\right) - \tan a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 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 80.9%

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

   1. tan-sum 99.6%

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

   2. div-inv 99.6%

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

4. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

6. Simplified 99.6%

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

Alternative 4: 79.5% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.9%

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

   1. tan-sum 99.6%

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

   2. div-inv 99.6%

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

4. Applied egg-rr 99.6%

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

   1. associate-*r/ 99.6%

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

   2. *-rgt-identity 99.6%

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

6. Simplified 99.6%

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

   1. expm1-log1p-u 94.1%

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

   2. expm1-undefine 94.2%

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

   3. log1p-undefine 94.2%

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

   4. add-exp-log 99.6%

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

8. Applied egg-rr 99.6%

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

   1. associate--l+ 99.6%

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

   2. fma-neg 99.6%

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

   3. metadata-eval 99.6%

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

10. Simplified 99.6%

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

    1. sub-neg 99.6%

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

    2. associate--r+ 99.7%

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

    3. metadata-eval 99.7%

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

12. Applied egg-rr 99.7%

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

    1. *-rgt-identity 99.7%

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

    2. associate-*r/ 99.6%

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

    3. fma-undefine 99.7%

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

    4. sub0-neg 99.7%

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

    5. fma-undefine 99.6%

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

    6. distribute-neg-in 99.6%

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

    7. metadata-eval 99.6%

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

    8. +-commutative 99.6%

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

    9. sub-neg 99.6%

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

    10. fma-neg 99.6%

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

14. Simplified 99.7%

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

15. Taylor expanded in y around 0 81.2%

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

16. Final simplification 81.2%

    \[\leadsto x + \left(\left(\tan y + \tan z\right) - \tan a\right) \]
17. Add Preprocessing

Alternative 5: 65.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 0.00039:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z a)
 :precision binary64
 (if (<= z 0.00039) (+ x (- (tan y) (tan a))) (+ x (tan (+ y z)))))

C

double code(double x, double y, double z, double a) {
	double tmp;
	if (z <= 0.00039) {
		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 (z <= 0.00039d0) 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 (z <= 0.00039) {
		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 z <= 0.00039:
		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 (z <= 0.00039)
		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 (z <= 0.00039)
		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[z, 0.00039], 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 z < 3.89999999999999993e-4

   1. Initial program 88.9%

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

      1. expm1-log1p-u 80.5%

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

      2. +-commutative 80.5%

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

      3. associate-+l- 80.5%

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

   4. Applied egg-rr 80.5%

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

   5. Taylor expanded in z around 0 71.9%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\frac{\sin y}{\cos y}} - \left(\tan a - x\right)\right)\right) \]
   6. Step-by-step derivation

      1. expm1-log1p-u 79.0%

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

      2. tan-quot 79.0%

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

      3. associate--r- 79.1%

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

   7. Applied egg-rr 79.1%

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

   if 3.89999999999999993e-4 < z

   1. Initial program 57.8%

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

      1. expm1-log1p-u 53.4%

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

      2. +-commutative 53.4%

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

      3. associate-+l- 53.4%

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

   4. Applied egg-rr 53.4%

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

   5. Taylor expanded in a around 0 37.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)\right)}\right) \]
   6. Step-by-step derivation

      1. log1p-define 37.0%

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

   7. Simplified 37.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}\right) \]
   8. Step-by-step derivation

      1. expm1-log1p-u 40.3%

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

      2. tan-quot 40.3%

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

      3. +-commutative 40.3%

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

   9. Applied egg-rr 40.3%

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

4. Final simplification 69.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 0.00039:\\ \;\;\;\;x + \left(\tan y - \tan a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \tan \left(y + z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 79.2% 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 80.9%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 7: 50.5% 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 80.9%

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

   1. expm1-log1p-u 73.5%

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

   2. +-commutative 73.5%

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

   3. associate-+l- 73.5%

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

4. Applied egg-rr 73.5%

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

5. Taylor expanded in a around 0 48.3%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\log \left(1 + \left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)\right)}\right) \]
6. Step-by-step derivation

   1. log1p-define 48.4%

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

7. Simplified 48.4%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(x + \frac{\sin \left(y + z\right)}{\cos \left(y + z\right)}\right)}\right) \]
8. Step-by-step derivation

   1. expm1-log1p-u 51.0%

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

   2. tan-quot 51.0%

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

   3. +-commutative 51.0%

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

9. Applied egg-rr 51.0%

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

10. Final simplification 51.0%

    \[\leadsto x + \tan \left(y + z\right) \]
11. Add Preprocessing

Alternative 8: 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 80.9%

   \[x + \left(\tan \left(y + z\right) - \tan a\right) \]
2. Add Preprocessing

3. Taylor expanded in x around inf 31.8%

   \[\leadsto \color{blue}{x} \]
4. Add Preprocessing

Alternative 9: 2.7% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x y z a) :precision binary64 -1.0)

C

double code(double x, double y, double z, double a) {
	return -1.0;
}

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 = -1.0d0
end function

Java

public static double code(double x, double y, double z, double a) {
	return -1.0;
}

Python

def code(x, y, z, a):
	return -1.0

Julia

function code(x, y, z, a)
	return -1.0
end

MATLAB

function tmp = code(x, y, z, a)
	tmp = -1.0;
end

Wolfram

code[x_, y_, z_, a_] := -1.0

Derivation

1. Initial program 80.9%

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

   1. expm1-log1p-u 73.5%

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

   2. +-commutative 73.5%

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

   3. associate-+l- 73.5%

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

4. Applied egg-rr 73.5%

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

5. Taylor expanded in x around inf 21.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{-1 \cdot \log \left(\frac{1}{x}\right)}\right) \]
6. Step-by-step derivation

   1. mul-1-neg 21.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{-\log \left(\frac{1}{x}\right)}\right) \]

   2. log-rec 21.0%

      \[\leadsto \mathsf{expm1}\left(-\color{blue}{\left(-\log x\right)}\right) \]

   3. remove-double-neg 21.0%

      \[\leadsto \mathsf{expm1}\left(\color{blue}{\log x}\right) \]

7. Simplified 21.0%

   \[\leadsto \mathsf{expm1}\left(\color{blue}{\log x}\right) \]

8. Taylor expanded in x around 0 3.6%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(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))))