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

Percentage Accurate: 77.2% → 93.1%

Time: 10.3s

Alternatives: 10

Speedup: 1.0×

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: 77.2% 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: 93.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x + y\right) + \frac{y \cdot \left(z - t\right)}{t - a}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \mathbf{elif}\;t\_1 \leq -5 \cdot 10^{-284}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{a \cdot \frac{z}{t} - \mathsf{fma}\left(a, \frac{a}{t}, a - z\right)}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 + \frac{z - t}{t - a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (+ x y) (/ (* y (- z t)) (- t a)))))
   (if (<= t_1 (- INFINITY))
     (fma y (/ (- z a) t) x)
     (if (<= t_1 -5e-284)
       t_1
       (if (<= t_1 0.0)
         (fma y (/ (- (* a (/ z t)) (fma a (/ a t) (- a z))) t) x)
         (fma y (+ 1.0 (/ (- z t) (- t a))) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x + y) + ((y * (z - t)) / (t - a));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(y, ((z - a) / t), x);
	} else if (t_1 <= -5e-284) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma(y, (((a * (z / t)) - fma(a, (a / t), (a - z))) / t), x);
	} else {
		tmp = fma(y, (1.0 + ((z - t) / (t - a))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x + y) + Float64(Float64(y * Float64(z - t)) / Float64(t - a)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(y, Float64(Float64(z - a) / t), x);
	elseif (t_1 <= -5e-284)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = fma(y, Float64(Float64(Float64(a * Float64(z / t)) - fma(a, Float64(a / t), Float64(a - z))) / t), x);
	else
		tmp = fma(y, Float64(1.0 + Float64(Float64(z - t) / Float64(t - a))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x + y), $MachinePrecision] + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, -5e-284], t$95$1, If[LessEqual[t$95$1, 0.0], N[(y * N[(N[(N[(a * N[(z / t), $MachinePrecision]), $MachinePrecision] - N[(a * N[(a / t), $MachinePrecision] + N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(1.0 + N[(N[(z - t), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -inf.0

   1. Initial program 32.8%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 78.1

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

   5. Simplified 78.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 78.1

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

   7. Applied egg-rr 78.1%

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

   if -inf.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -4.99999999999999973e-284

   1. Initial program 97.8%

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

   if -4.99999999999999973e-284 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 3.9%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 46.3

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

   5. Simplified 46.3%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{\frac{\frac{a \cdot z}{t} - \left(a + \left(-1 \cdot z + \frac{{a}^{2}}{t}\right)\right)}{t}}, x\right) \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. associate-+r+ N/A

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

      7. +-commutative N/A

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

      8. unpow2 N/A

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

      9. associate-/l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. mul-1-neg N/A

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

      13. sub-neg N/A

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

      14. lower--.f64 99.7

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

   8. Simplified 99.7%

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

   if 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 82.1%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 92.0

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

   5. Simplified 92.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) + \frac{y \cdot \left(z - t\right)}{t - a} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \mathbf{elif}\;\left(x + y\right) + \frac{y \cdot \left(z - t\right)}{t - a} \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\left(x + y\right) + \frac{y \cdot \left(z - t\right)}{t - a}\\ \mathbf{elif}\;\left(x + y\right) + \frac{y \cdot \left(z - t\right)}{t - a} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{a \cdot \frac{z}{t} - \mathsf{fma}\left(a, \frac{a}{t}, a - z\right)}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 + \frac{z - t}{t - a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 93.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ t_2 := \left(x + y\right) + \frac{y \cdot \left(z - t\right)}{t - a}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-284}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 + \frac{z - t}{t - a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ (- z a) t) x))
        (t_2 (+ (+ x y) (/ (* y (- z t)) (- t a)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -5e-284)
       t_2
       (if (<= t_2 0.0) t_1 (fma y (+ 1.0 (/ (- z t) (- t a))) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, ((z - a) / t), x);
	double t_2 = (x + y) + ((y * (z - t)) / (t - a));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -5e-284) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else {
		tmp = fma(y, (1.0 + ((z - t) / (t - a))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(Float64(z - a) / t), x)
	t_2 = Float64(Float64(x + y) + Float64(Float64(y * Float64(z - t)) / Float64(t - a)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -5e-284)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	else
		tmp = fma(y, Float64(1.0 + Float64(Float64(z - t) / Float64(t - a))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -5e-284], t$95$2, If[LessEqual[t$95$2, 0.0], t$95$1, N[(y * N[(1.0 + N[(N[(z - t), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -inf.0 or -4.99999999999999973e-284 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 19.7%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 86.1

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

   5. Simplified 86.1%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 87.9

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

   7. Applied egg-rr 87.9%

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

   if -inf.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -4.99999999999999973e-284

   1. Initial program 97.8%

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

   if 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 82.1%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 92.0

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

   5. Simplified 92.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) + \frac{y \cdot \left(z - t\right)}{t - a} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \mathbf{elif}\;\left(x + y\right) + \frac{y \cdot \left(z - t\right)}{t - a} \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\left(x + y\right) + \frac{y \cdot \left(z - t\right)}{t - a}\\ \mathbf{elif}\;\left(x + y\right) + \frac{y \cdot \left(z - t\right)}{t - a} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 + \frac{z - t}{t - a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 + \frac{z - t}{t - a}, x\right)\\ t_2 := \left(x + y\right) + \frac{y \cdot \left(z - t\right)}{t - a}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-284}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (+ 1.0 (/ (- z t) (- t a))) x))
        (t_2 (+ (+ x y) (/ (* y (- z t)) (- t a)))))
   (if (<= t_2 -5e-284) t_1 (if (<= t_2 0.0) (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 - t) / (t - a))), x);
	double t_2 = (x + y) + ((y * (z - t)) / (t - a));
	double tmp;
	if (t_2 <= -5e-284) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		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(Float64(z - t) / Float64(t - a))), x)
	t_2 = Float64(Float64(x + y) + Float64(Float64(y * Float64(z - t)) / Float64(t - a)))
	tmp = 0.0
	if (t_2 <= -5e-284)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		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[(N[(z - t), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + y), $MachinePrecision] + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-284], t$95$1, If[LessEqual[t$95$2, 0.0], N[(y * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < -4.99999999999999973e-284 or 0.0 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t)))

   1. Initial program 82.6%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 90.7

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

   5. Simplified 90.7%

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

   if -4.99999999999999973e-284 < (-.f64 (+.f64 x y) (/.f64 (*.f64 (-.f64 z t) y) (-.f64 a t))) < 0.0

   1. Initial program 3.9%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 95.8

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

   5. Simplified 95.8%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 99.7

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

   7. Applied egg-rr 99.7%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x + y\right) + \frac{y \cdot \left(z - t\right)}{t - a} \leq -5 \cdot 10^{-284}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 + \frac{z - t}{t - a}, x\right)\\ \mathbf{elif}\;\left(x + y\right) + \frac{y \cdot \left(z - t\right)}{t - a} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - a}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 + \frac{z - t}{t - a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -2 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t}, z - a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ z a)) x)))
   (if (<= a -2e-9) t_1 (if (<= a 6.2e-110) (fma (/ y t) (- z a) 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 <= -2e-9) {
		tmp = t_1;
	} else if (a <= 6.2e-110) {
		tmp = fma((y / t), (z - a), 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 <= -2e-9)
		tmp = t_1;
	elseif (a <= 6.2e-110)
		tmp = fma(Float64(y / t), Float64(z - a), 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, -2e-9], t$95$1, If[LessEqual[a, 6.2e-110], N[(N[(y / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.00000000000000012e-9 or 6.20000000000000014e-110 < a

   1. Initial program 81.0%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 86.3

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

   5. Simplified 86.3%

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

   if -2.00000000000000012e-9 < a < 6.20000000000000014e-110

   1. Initial program 67.5%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 86.2

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

   5. Simplified 86.2%

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

Alternative 5: 81.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{-43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-110}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ z a)) x)))
   (if (<= a -2.3e-43) t_1 (if (<= a 6.2e-110) (fma z (/ y 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 <= -2.3e-43) {
		tmp = t_1;
	} else if (a <= 6.2e-110) {
		tmp = fma(z, (y / 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 <= -2.3e-43)
		tmp = t_1;
	elseif (a <= 6.2e-110)
		tmp = fma(z, Float64(y / 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, -2.3e-43], t$95$1, If[LessEqual[a, 6.2e-110], N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.2999999999999999e-43 or 6.20000000000000014e-110 < a

   1. Initial program 79.9%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 85.4

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

   5. Simplified 85.4%

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

   if -2.2999999999999999e-43 < a < 6.20000000000000014e-110

   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 x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 77.8

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

   5. Simplified 77.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 63.9

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

   7. Applied egg-rr 63.9%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 84.6

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

   10. Simplified 84.6%

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

Alternative 6: 77.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-7}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 0.0076:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.3e-7) (+ x y) (if (<= a 0.0076) (fma z (/ y t) x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.3e-7) {
		tmp = x + y;
	} else if (a <= 0.0076) {
		tmp = fma(z, (y / t), x);
	} else {
		tmp = x + y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.3e-7)
		tmp = Float64(x + y);
	elseif (a <= 0.0076)
		tmp = fma(z, Float64(y / t), x);
	else
		tmp = Float64(x + y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.3e-7], N[(x + y), $MachinePrecision], If[LessEqual[a, 0.0076], N[(z * N[(y / t), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.29999999999999999e-7 or 0.00759999999999999998 < a

   1. Initial program 81.5%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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. Simplified 75.4%

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

   if -1.29999999999999999e-7 < a < 0.00759999999999999998

   1. Initial program 69.3%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 79.1

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

   5. Simplified 79.1%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

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

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 66.5

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

   7. Applied egg-rr 66.5%

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

   8. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 80.6

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

   10. Simplified 80.6%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-7}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 0.0076:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-7}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.3e-7) (+ x y) (if (<= a 2.1e-18) (fma y (/ z t) x) (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.3e-7) {
		tmp = x + y;
	} else if (a <= 2.1e-18) {
		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 <= -1.3e-7)
		tmp = Float64(x + y);
	elseif (a <= 2.1e-18)
		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, -1.3e-7], N[(x + y), $MachinePrecision], If[LessEqual[a, 2.1e-18], N[(y * N[(z / t), $MachinePrecision] + x), $MachinePrecision], N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.29999999999999999e-7 or 2.1e-18 < a

   1. Initial program 81.5%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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. Simplified 75.4%

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

   if -1.29999999999999999e-7 < a < 2.1e-18

   1. Initial program 69.3%

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

   3. Taylor expanded in x around 0

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

      3. *-rgt-identity N/A

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

      4. associate-/l* N/A

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

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

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

      6. unsub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 79.1

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

   5. Simplified 79.1%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 76.9

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

   8. Simplified 76.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.3 \cdot 10^{-7}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 2.1 \cdot 10^{-18}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 63.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-43}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-26}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -5.5e-43) (+ x y) (if (<= a 4.4e-26) x (+ x y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -5.5e-43) {
		tmp = x + y;
	} else if (a <= 4.4e-26) {
		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 <= (-5.5d-43)) then
        tmp = x + y
    else if (a <= 4.4d-26) 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 <= -5.5e-43) {
		tmp = x + y;
	} else if (a <= 4.4e-26) {
		tmp = x;
	} else {
		tmp = x + y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -5.5e-43:
		tmp = x + y
	elif a <= 4.4e-26:
		tmp = x
	else:
		tmp = x + y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -5.5e-43)
		tmp = Float64(x + y);
	elseif (a <= 4.4e-26)
		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 <= -5.5e-43)
		tmp = x + y;
	elseif (a <= 4.4e-26)
		tmp = x;
	else
		tmp = x + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -5.5e-43], N[(x + y), $MachinePrecision], If[LessEqual[a, 4.4e-26], x, N[(x + y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.50000000000000013e-43 or 4.4000000000000002e-26 < a

   1. Initial program 80.4%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 74.7

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

   5. Simplified 74.7%

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

   if -5.50000000000000013e-43 < a < 4.4000000000000002e-26

   1. Initial program 69.6%

      \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
   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 53.3

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

   5. Simplified 53.3%

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

   6. Taylor expanded in z around 0

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

      1. distribute-rgt1-in N/A

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

      2. metadata-eval N/A

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

      3. mul0-lft N/A

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

      4. lower-+.f64 52.0

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

   8. Simplified 52.0%

      \[\leadsto \color{blue}{x + 0} \]
   9. Step-by-step derivation

      1. +-rgt-identity 52.0

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

   10. Applied egg-rr 52.0%

       \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-43}:\\ \;\;\;\;x + y\\ \mathbf{elif}\;a \leq 4.4 \cdot 10^{-26}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.6% 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 75.2%

   \[\left(x + y\right) - \frac{\left(z - t\right) \cdot y}{a - t} \]
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 48.0

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

5. Simplified 48.0%

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

6. Taylor expanded in z around 0

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

   1. distribute-rgt1-in N/A

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

   2. metadata-eval N/A

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

   3. mul0-lft N/A

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

   4. lower-+.f64 51.2

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

8. Simplified 51.2%

   \[\leadsto \color{blue}{x + 0} \]
9. Step-by-step derivation

   1. +-rgt-identity 51.2

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

10. Applied egg-rr 51.2%

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

Alternative 10: 2.7% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 75.2%

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

3. Taylor expanded in x around 0

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

   1. associate--l+ N/A

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

   2. +-commutative N/A

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

   3. *-rgt-identity N/A

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

   4. associate-/l* N/A

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

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

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

   6. unsub-neg N/A

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

   7. mul-1-neg N/A

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

   8. lower-fma.f64 N/A

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

   9. mul-1-neg N/A

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

   10. unsub-neg N/A

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

   11. lower--.f64 N/A

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

   12. lower-/.f64 N/A

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

   13. lower--.f64 N/A

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

   14. lower--.f64 86.5

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

5. Simplified 86.5%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. sub-neg N/A

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

   5. +-commutative N/A

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

   6. lift-/.f64 N/A

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

   7. div-inv N/A

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

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

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

   9. lower-fma.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-/.f64 78.2

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

7. Applied egg-rr 78.2%

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

8. Taylor expanded in y around inf

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f64 N/A

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

   7. sub-neg N/A

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

   8. mul-1-neg N/A

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

   9. +-commutative N/A

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

   10. distribute-lft-in N/A

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

   11. neg-mul-1 N/A

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

   12. mul-1-neg N/A

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

   13. remove-double-neg N/A

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

   14. mul-1-neg N/A

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

   15. sub-neg N/A

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

   16. lower--.f64 N/A

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

   17. lower--.f64 38.1

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

10. Simplified 38.1%

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

11. Taylor expanded in t around inf

    \[\leadsto \color{blue}{y + -1 \cdot y} \]
12. Step-by-step derivation

    1. distribute-rgt1-in N/A

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

    2. metadata-eval N/A

       \[\leadsto \color{blue}{0} \cdot y \]

    3. mul0-lft 2.6

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

13. Simplified 2.6%

    \[\leadsto \color{blue}{0} \]
14. Add Preprocessing

Developer Target 1: 88.6% 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 2024207 
(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))))