Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B

Percentage Accurate: 77.4% → 88.8%

Time: 9.8s

Alternatives: 10

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y, double z, double t, double a) {
	return (x + y) - (((z - t) * y) / (a - t));
}

Python

def code(x, y, z, t, a):
	return (x + y) - (((z - t) * y) / (a - t))

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 77.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y, double z, double t, double a) {
	return (x + y) - (((z - t) * y) / (a - t));
}

Python

def code(x, y, z, t, a):
	return (x + y) - (((z - t) * y) / (a - t))

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{if}\;t \leq -5.1 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(t - z\right), \frac{-1}{t - a}, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y t) (- z a) x)))
   (if (<= t -5.1e+88)
     t_1
     (if (<= t 4.2e+47) (fma (* y (- t z)) (/ -1.0 (- t a)) (+ x y)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / t), (z - a), x);
	double tmp;
	if (t <= -5.1e+88) {
		tmp = t_1;
	} else if (t <= 4.2e+47) {
		tmp = fma((y * (t - z)), (-1.0 / (t - a)), (x + y));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / t), Float64(z - a), x)
	tmp = 0.0
	if (t <= -5.1e+88)
		tmp = t_1;
	elseif (t <= 4.2e+47)
		tmp = fma(Float64(y * Float64(t - z)), Float64(-1.0 / Float64(t - a)), Float64(x + y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -5.1e+88], t$95$1, If[LessEqual[t, 4.2e+47], N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(t - a), $MachinePrecision]), $MachinePrecision] + N[(x + y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.0999999999999997e88 or 4.2e47 < t

   1. Initial program 57.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. mul-1-neg N/A

         \[\leadsto \left(\frac{y \cdot z}{t} + \color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot y}{t}\right)\right)}\right) + x \]

      8. sub-neg N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out-- N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if -5.0999999999999997e88 < t < 4.2e47

   1. Initial program 90.0%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. div-inv N/A

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

      6. distribute-lft-neg-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. distribute-rgt-neg-in N/A

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

      10. lower-*.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 90.0

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

   4. Applied rewrites 90.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.1 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+47}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(t - z\right), \frac{-1}{t - a}, x + y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 88.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{if}\;t \leq -5.1 \cdot 10^{+88}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+47}:\\ \;\;\;\;\left(x + y\right) + \frac{y \cdot \left(t - z\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y t) (- z a) x)))
   (if (<= t -5.1e+88)
     t_1
     (if (<= t 4.2e+47) (+ (+ x y) (/ (* y (- t z)) (- a t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / t), (z - a), x);
	double tmp;
	if (t <= -5.1e+88) {
		tmp = t_1;
	} else if (t <= 4.2e+47) {
		tmp = (x + y) + ((y * (t - z)) / (a - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / t), Float64(z - a), x)
	tmp = 0.0
	if (t <= -5.1e+88)
		tmp = t_1;
	elseif (t <= 4.2e+47)
		tmp = Float64(Float64(x + y) + Float64(Float64(y * Float64(t - z)) / Float64(a - t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -5.1e+88], t$95$1, If[LessEqual[t, 4.2e+47], N[(N[(x + y), $MachinePrecision] + N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.0999999999999997e88 or 4.2e47 < t

   1. Initial program 57.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. mul-1-neg N/A

         \[\leadsto \left(\frac{y \cdot z}{t} + \color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot y}{t}\right)\right)}\right) + x \]

      8. sub-neg N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out-- N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 91.0

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

   5. Applied rewrites 91.0%

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

   if -5.0999999999999997e88 < t < 4.2e47

   1. Initial program 90.0%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.1 \cdot 10^{+88}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+47}:\\ \;\;\;\;\left(x + y\right) + \frac{y \cdot \left(t - z\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + y\right) - z \cdot \frac{y}{a}\\ \mathbf{if}\;a \leq -6.1 \cdot 10^{-108}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ x y) (* z (/ y a)))))
   (if (<= a -6.1e-108) t_1 (if (<= a 4e+82) (fma y (/ (- z a) t) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x + y) - (z * (y / a));
	double tmp;
	if (a <= -6.1e-108) {
		tmp = t_1;
	} else if (a <= 4e+82) {
		tmp = fma(y, ((z - a) / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x + y) - Float64(z * Float64(y / a)))
	tmp = 0.0
	if (a <= -6.1e-108)
		tmp = t_1;
	elseif (a <= 4e+82)
		tmp = fma(y, Float64(Float64(z - a) / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] - N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6.1e-108], t$95$1, If[LessEqual[a, 4e+82], N[(y * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.10000000000000007e-108 or 3.9999999999999999e82 < a

   1. Initial program 81.9%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 84.9

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

   5. Applied rewrites 84.9%

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

   if -6.10000000000000007e-108 < a < 3.9999999999999999e82

   1. Initial program 70.3%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. mul-1-neg N/A

         \[\leadsto \left(\frac{y \cdot z}{t} + \color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot y}{t}\right)\right)}\right) + x \]

      8. sub-neg N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out-- N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 86.6

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 87.2%

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

Alternative 4: 82.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{if}\;t \leq -2.9 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.2 \cdot 10^{-76}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y t) (- z a) x)))
   (if (<= t -2.9e-5) t_1 (if (<= t 6.2e-76) (fma y (- 1.0 (/ z a)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / t), (z - a), x);
	double tmp;
	if (t <= -2.9e-5) {
		tmp = t_1;
	} else if (t <= 6.2e-76) {
		tmp = fma(y, (1.0 - (z / a)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / t), Float64(z - a), x)
	tmp = 0.0
	if (t <= -2.9e-5)
		tmp = t_1;
	elseif (t <= 6.2e-76)
		tmp = fma(y, Float64(1.0 - Float64(z / a)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -2.9e-5], t$95$1, If[LessEqual[t, 6.2e-76], N[(y * N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.9e-5 or 6.19999999999999939e-76 < t

   1. Initial program 65.1%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. mul-1-neg N/A

         \[\leadsto \left(\frac{y \cdot z}{t} + \color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot y}{t}\right)\right)}\right) + x \]

      8. sub-neg N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out-- N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 86.7

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

   5. Applied rewrites 86.7%

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

   if -2.9e-5 < t < 6.19999999999999939e-76

   1. Initial program 92.7%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-out-- N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 85.1

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

   5. Applied rewrites 85.1%

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

Alternative 5: 81.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -6.1 \cdot 10^{-108}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ z a)) x)))
   (if (<= a -6.1e-108) t_1 (if (<= a 2.9e+82) (fma y (/ (- z a) t) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (1.0 - (z / a)), x);
	double tmp;
	if (a <= -6.1e-108) {
		tmp = t_1;
	} else if (a <= 2.9e+82) {
		tmp = fma(y, ((z - a) / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(1.0 - Float64(z / a)), x)
	tmp = 0.0
	if (a <= -6.1e-108)
		tmp = t_1;
	elseif (a <= 2.9e+82)
		tmp = fma(y, Float64(Float64(z - a) / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -6.1e-108], t$95$1, If[LessEqual[a, 2.9e+82], N[(y * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.10000000000000007e-108 or 2.9000000000000001e82 < a

   1. Initial program 81.9%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-out-- N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

   if -6.10000000000000007e-108 < a < 2.9000000000000001e82

   1. Initial program 70.3%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. mul-1-neg N/A

         \[\leadsto \left(\frac{y \cdot z}{t} + \color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot y}{t}\right)\right)}\right) + x \]

      8. sub-neg N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out-- N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 86.6

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

   5. Applied rewrites 86.6%

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

   7. Applied rewrites 87.2%

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

Alternative 6: 80.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -6.1 \cdot 10^{-108}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.9 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ z a)) x)))
   (if (<= a -6.1e-108) t_1 (if (<= a 2.9e+82) (fma y (/ z t) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (1.0 - (z / a)), x);
	double tmp;
	if (a <= -6.1e-108) {
		tmp = t_1;
	} else if (a <= 2.9e+82) {
		tmp = fma(y, (z / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(1.0 - Float64(z / a)), x)
	tmp = 0.0
	if (a <= -6.1e-108)
		tmp = t_1;
	elseif (a <= 2.9e+82)
		tmp = fma(y, Float64(z / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(z / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -6.1e-108], t$95$1, If[LessEqual[a, 2.9e+82], N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.10000000000000007e-108 or 2.9000000000000001e82 < a

   1. Initial program 81.9%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-out-- N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-/.f64 84.8

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

   5. Applied rewrites 84.8%

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

   if -6.10000000000000007e-108 < a < 2.9000000000000001e82

   1. Initial program 70.3%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 39.2

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

   5. Applied rewrites 39.2%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto \left(x + y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(z - t\right)}{t}\right)\right)}\right)\right) \]

      3. remove-double-neg N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

         \[\leadsto \left(x + y\right) + y \cdot \color{blue}{\left(\frac{z}{t} + \left(\mathsf{neg}\left(\frac{t}{t}\right)\right)\right)} \]

      7. *-inverses N/A

         \[\leadsto \left(x + y\right) + y \cdot \left(\frac{z}{t} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

   8. Applied rewrites 82.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{z}{t}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 77.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{+25}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.65e+25) (+ x y) (if (<= a 4e+82) (fma y (/ z t) x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.65e+25) {
		tmp = x + y;
	} else if (a <= 4e+82) {
		tmp = fma(y, (z / t), x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.65e+25)
		tmp = Float64(x + y);
	elseif (a <= 4e+82)
		tmp = fma(y, Float64(z / t), x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.65e+25], N[(x + y), $MachinePrecision], If[LessEqual[a, 4e+82], N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6500000000000001e25 or 3.9999999999999999e82 < a

   1. Initial program 81.1%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 78.9

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

   5. Applied rewrites 78.9%

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

   if -1.6500000000000001e25 < a < 3.9999999999999999e82

   1. Initial program 73.2%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 41.5

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

   5. Applied rewrites 41.5%

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

   6. Taylor expanded in a around 0

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

      1. sub-neg N/A

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

      2. mul-1-neg N/A

         \[\leadsto \left(x + y\right) + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot \left(z - t\right)}{t}\right)\right)}\right)\right) \]

      3. remove-double-neg N/A

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

      4. associate-/l* N/A

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

      5. div-sub N/A

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

      6. sub-neg N/A

         \[\leadsto \left(x + y\right) + y \cdot \color{blue}{\left(\frac{z}{t} + \left(\mathsf{neg}\left(\frac{t}{t}\right)\right)\right)} \]

      7. *-inverses N/A

         \[\leadsto \left(x + y\right) + y \cdot \left(\frac{z}{t} + \left(\mathsf{neg}\left(\color{blue}{1}\right)\right)\right) \]

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-out N/A

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

      11. *-commutative N/A

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

      12. associate-/l* N/A

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

      13. associate-+r+ N/A

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

      14. +-commutative N/A

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

   8. Applied rewrites 76.2%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.65 \cdot 10^{+25}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 61.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{-237}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-195}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{elif}\;a \leq 2.85 \cdot 10^{+82}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -3.3e-237)
   (+ x y)
   (if (<= a 4.4e-195) (/ (* y z) t) (if (<= a 2.85e+82) x (+ x y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -3.3e-237) {
		tmp = x + y;
	} else if (a <= 4.4e-195) {
		tmp = (y * z) / t;
	} else if (a <= 2.85e+82) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-3.3d-237)) then
        tmp = x + y
    else if (a <= 4.4d-195) then
        tmp = (y * z) / t
    else if (a <= 2.85d+82) then
        tmp = x
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -3.3e-237) {
		tmp = x + y;
	} else if (a <= 4.4e-195) {
		tmp = (y * z) / t;
	} else if (a <= 2.85e+82) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -3.3e-237:
		tmp = x + y
	elif a <= 4.4e-195:
		tmp = (y * z) / t
	elif a <= 2.85e+82:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -3.3e-237)
		tmp = Float64(x + y);
	elseif (a <= 4.4e-195)
		tmp = Float64(Float64(y * z) / t);
	elseif (a <= 2.85e+82)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -3.3e-237)
		tmp = x + y;
	elseif (a <= 4.4e-195)
		tmp = (y * z) / t;
	elseif (a <= 2.85e+82)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -3.3e-237], N[(x + y), $MachinePrecision], If[LessEqual[a, 4.4e-195], N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[a, 2.85e+82], x, N[(x + y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.3000000000000001e-237 or 2.85000000000000008e82 < a

   1. Initial program 81.9%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 67.2

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

   5. Applied rewrites 67.2%

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

   if -3.3000000000000001e-237 < a < 4.40000000000000011e-195

   1. Initial program 68.4%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. mul-1-neg N/A

         \[\leadsto \left(\frac{y \cdot z}{t} + \color{blue}{\left(\mathsf{neg}\left(\frac{a \cdot y}{t}\right)\right)}\right) + x \]

      8. sub-neg N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. associate-/l* N/A

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

      12. distribute-rgt-out-- N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 90.8

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

   5. Applied rewrites 90.8%

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

   6. Taylor expanded in z around inf

      \[\leadsto \frac{y \cdot z}{\color{blue}{t}} \]
   7. Step-by-step derivation

   8. Applied rewrites 64.8%

      \[\leadsto \frac{y \cdot z}{\color{blue}{t}} \]

   if 4.40000000000000011e-195 < a < 2.85000000000000008e82

   1. Initial program 62.5%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 63.2

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

   5. Applied rewrites 63.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 63.3%

      \[\leadsto x + \color{blue}{0} \]

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 63.3%

       \[\leadsto x \]
3. Recombined 3 regimes into one program.

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.3 \cdot 10^{-237}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-195}:\\ \;\;\;\;\frac{y \cdot z}{t}\\ \mathbf{elif}\;a \leq 2.85 \cdot 10^{+82}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{-39}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 2.85 \cdot 10^{+82}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -2.05e-39) (+ x y) (if (<= a 2.85e+82) x (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.05e-39) {
		tmp = x + y;
	} else if (a <= 2.85e+82) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (a <= (-2.05d-39)) then
        tmp = x + y
    else if (a <= 2.85d+82) then
        tmp = x
    else
        tmp = x + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -2.05e-39) {
		tmp = x + y;
	} else if (a <= 2.85e+82) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -2.05e-39:
		tmp = x + y
	elif a <= 2.85e+82:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -2.05e-39)
		tmp = Float64(x + y);
	elseif (a <= 2.85e+82)
		tmp = x;
	else
		tmp = Float64(x + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -2.05e-39)
		tmp = x + y;
	elseif (a <= 2.85e+82)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -2.05e-39], N[(x + y), $MachinePrecision], If[LessEqual[a, 2.85e+82], x, N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.05e-39 or 2.85000000000000008e82 < a

   1. Initial program 81.3%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

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

      2. lower-+.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if -2.05e-39 < a < 2.85000000000000008e82

   1. Initial program 72.1%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   2. Add Preprocessing

   3. Taylor expanded in a around 0

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

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 64.2

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

   5. Applied rewrites 64.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 48.7%

      \[\leadsto x + \color{blue}{0} \]

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 48.7%

       \[\leadsto x \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.05 \cdot 10^{-39}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 2.85 \cdot 10^{+82}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 51.2% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 76.3%

   \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
2. Add Preprocessing

3. Taylor expanded in a around 0

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

   1. cancel-sign-sub-inv N/A

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

   2. metadata-eval N/A

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

   3. *-lft-identity N/A

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

   4. +-commutative N/A

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

   5. associate-/l* N/A

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

   6. lower-fma.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. +-commutative N/A

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

   10. lower-+.f64 54.0

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

5. Applied rewrites 54.0%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 51.1%

   \[\leadsto x + \color{blue}{0} \]

9. Taylor expanded in z around 0

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

11. Applied rewrites 51.1%

    \[\leadsto x \]
12. Add Preprocessing

Developer Target 1: 88.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + x\right) - \left(\left(z - t\right) \cdot \frac{1}{a - t}\right) \cdot y\\ t_2 := \left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t}\\ \mathbf{if}\;t\_2 < -1.3664970889390727 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.4754293444577233 \cdot 10^{-239}:\\ \;\;\;\;\frac{y \cdot \left(a - z\right) - x \cdot t}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (+ y x) (* (* (- z t) (/ 1.0 (- a t))) y)))
        (t_2 (- (+ x y) (/ (* (- z t) y) (- a t)))))
   (if (< t_2 -1.3664970889390727e-7)
     t_1
     (if (< t_2 1.4754293444577233e-239)
       (/ (- (* y (- a z)) (* x t)) (- a t))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + x) - (((z - t) * (1.0 / (a - t))) * y);
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 < -1.3664970889390727e-7) {
		tmp = t_1;
	} else if (t_2 < 1.4754293444577233e-239) {
		tmp = ((y * (a - z)) - (x * t)) / (a - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (y + x) - (((z - t) * (1.0d0 / (a - t))) * y)
    t_2 = (x + y) - (((z - t) * y) / (a - t))
    if (t_2 < (-1.3664970889390727d-7)) then
        tmp = t_1
    else if (t_2 < 1.4754293444577233d-239) then
        tmp = ((y * (a - z)) - (x * t)) / (a - t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y + x) - (((z - t) * (1.0 / (a - t))) * y);
	double t_2 = (x + y) - (((z - t) * y) / (a - t));
	double tmp;
	if (t_2 < -1.3664970889390727e-7) {
		tmp = t_1;
	} else if (t_2 < 1.4754293444577233e-239) {
		tmp = ((y * (a - z)) - (x * t)) / (a - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y + x) - (((z - t) * (1.0 / (a - t))) * y)
	t_2 = (x + y) - (((z - t) * y) / (a - t))
	tmp = 0
	if t_2 < -1.3664970889390727e-7:
		tmp = t_1
	elif t_2 < 1.4754293444577233e-239:
		tmp = ((y * (a - z)) - (x * t)) / (a - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y + x) - Float64(Float64(Float64(z - t) * Float64(1.0 / Float64(a - t))) * y))
	t_2 = Float64(Float64(x + y) - Float64(Float64(Float64(z - t) * y) / Float64(a - t)))
	tmp = 0.0
	if (t_2 < -1.3664970889390727e-7)
		tmp = t_1;
	elseif (t_2 < 1.4754293444577233e-239)
		tmp = Float64(Float64(Float64(y * Float64(a - z)) - Float64(x * t)) / Float64(a - t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y + x) - (((z - t) * (1.0 / (a - t))) * y);
	t_2 = (x + y) - (((z - t) * y) / (a - t));
	tmp = 0.0;
	if (t_2 < -1.3664970889390727e-7)
		tmp = t_1;
	elseif (t_2 < 1.4754293444577233e-239)
		tmp = ((y * (a - z)) - (x * t)) / (a - t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y + x), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] - N[(N[(N[(z - t), $MachinePrecision] * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -1.3664970889390727e-7], t$95$1, If[Less[t$95$2, 1.4754293444577233e-239], N[(N[(N[(y * N[(a - z), $MachinePrecision]), $MachinePrecision] - N[(x * t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024222 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) -13664970889390727/100000000000000000000000) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)) (if (< (- (+ x y) (/ (* (- z t) y) (- a t))) 14754293444577233/1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (- (* y (- a z)) (* x t)) (- a t)) (- (+ y x) (* (* (- z t) (/ 1 (- a t))) y)))))

  (- (+ x y) (/ (* (- z t) y) (- a t))))