Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3

Percentage Accurate: 97.1% → 99.5%

Time: 9.6s

Alternatives: 13

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

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 - z) + 1.0d0) / a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - z) + 1.0d0) / a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - (a / (((-1.0d0) - (t - z)) / (z - y)))
end function

Java

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

Python

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

Julia

function code(x, y, z, t, a)
	return Float64(x - Float64(a / Float64(Float64(-1.0 - Float64(t - z)) / Float64(z - y))))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.2%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. *-commutative N/A

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

   5. clear-num N/A

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

   6. un-div-inv N/A

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

   7. lower-/.f64 N/A

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

   8. lower-/.f64 99.5

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lower-+.f64 99.5

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

4. Applied rewrites 99.5%

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

5. Final simplification 99.5%

   \[\leadsto x - \frac{a}{\frac{-1 - \left(t - z\right)}{z - y}} \]
6. Add Preprocessing

Alternative 2: 73.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y \cdot a}{t}\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+216}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, a, x\right)\\ \mathbf{elif}\;t \leq -3.8 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 58:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{z - 1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* y a) t))))
   (if (<= t -6.5e+216)
     (fma (/ z t) a x)
     (if (<= t -3.8e+41) t_1 (if (<= t 58.0) (fma a (/ y (- z 1.0)) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y * a) / t);
	double tmp;
	if (t <= -6.5e+216) {
		tmp = fma((z / t), a, x);
	} else if (t <= -3.8e+41) {
		tmp = t_1;
	} else if (t <= 58.0) {
		tmp = fma(a, (y / (z - 1.0)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y * a) / t))
	tmp = 0.0
	if (t <= -6.5e+216)
		tmp = fma(Float64(z / t), a, x);
	elseif (t <= -3.8e+41)
		tmp = t_1;
	elseif (t <= 58.0)
		tmp = fma(a, Float64(y / Float64(z - 1.0)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y * a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -6.5e+216], N[(N[(z / t), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[t, -3.8e+41], t$95$1, If[LessEqual[t, 58.0], N[(a * N[(y / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.50000000000000029e216

   1. Initial program 96.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. metadata-eval N/A

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

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

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-+.f64 90.3

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

   5. Applied rewrites 90.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 82.9%

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

   if -6.50000000000000029e216 < t < -3.8000000000000001e41 or 58 < t

   1. Initial program 98.3%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 80.8

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

   5. Applied rewrites 80.8%

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

   6. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 80.8%

      \[\leadsto x - \frac{y \cdot a}{t} \]

   if -3.8000000000000001e41 < t < 58

   1. Initial program 96.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. clear-num-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 75.1%

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

Alternative 3: 91.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{y - z}{t}, x\right)\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z - y}{1 - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{a}{-t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -5e+44)
   (fma (- a) (/ (- y z) t) x)
   (if (<= t 8.2e+76)
     (fma a (/ (- z y) (- 1.0 z)) x)
     (fma (- y z) (/ a (- t)) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -5e+44) {
		tmp = fma(-a, ((y - z) / t), x);
	} else if (t <= 8.2e+76) {
		tmp = fma(a, ((z - y) / (1.0 - z)), x);
	} else {
		tmp = fma((y - z), (a / -t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -5e+44)
		tmp = fma(Float64(-a), Float64(Float64(y - z) / t), x);
	elseif (t <= 8.2e+76)
		tmp = fma(a, Float64(Float64(z - y) / Float64(1.0 - z)), x);
	else
		tmp = fma(Float64(y - z), Float64(a / Float64(-t)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -5e+44], N[((-a) * N[(N[(y - z), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 8.2e+76], N[(a * N[(N[(z - y), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(a / (-t)), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.9999999999999996e44

   1. Initial program 96.6%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. associate-*r/ N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

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

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower--.f64 88.4

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

   5. Applied rewrites 88.4%

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

   if -4.9999999999999996e44 < t < 8.1999999999999997e76

   1. Initial program 96.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. clear-num-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 96.9

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 97.3

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

   7. Applied rewrites 97.3%

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

   if 8.1999999999999997e76 < t

   1. Initial program 98.7%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. clear-num-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 98.7

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

   4. Applied rewrites 98.7%

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

   5. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.8

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

   7. Applied rewrites 90.8%

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

4. Final simplification 93.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(-a, \frac{y - z}{t}, x\right)\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+76}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z - y}{1 - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{a}{-t}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{a}{z}, x\right)\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{+85}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 128000:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ a z) x)))
   (if (<= z -3.2e+85)
     t_1
     (if (<= z 128000.0) (fma a (/ y (- -1.0 t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), (a / z), x);
	double tmp;
	if (z <= -3.2e+85) {
		tmp = t_1;
	} else if (z <= 128000.0) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(a / z), x)
	tmp = 0.0
	if (z <= -3.2e+85)
		tmp = t_1;
	elseif (z <= 128000.0)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(a / z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -3.2e+85], t$95$1, If[LessEqual[z, 128000.0], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.20000000000000018e85 or 128000 < z

   1. Initial program 94.6%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. clear-num-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 94.7

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

   4. Applied rewrites 94.7%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 85.2

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

   7. Applied rewrites 85.2%

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

   if -3.20000000000000018e85 < z < 128000

   1. Initial program 99.2%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower--.f64 45.3

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

   5. Applied rewrites 45.3%

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

   6. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{a} \]
   7. Step-by-step derivation

   8. Applied rewrites 4.7%

      \[\leadsto -a \]

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x - \frac{a \cdot y}{1 + t}} \]
   10. Step-by-step derivation

       1. sub-neg N/A

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

       2. mul-1-neg N/A

          \[\leadsto x + \color{blue}{-1 \cdot \frac{a \cdot y}{1 + t}} \]

       3. +-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot y}{1 + t} + x} \]

       4. mul-1-neg N/A

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

       5. associate-/l* N/A

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

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

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

       7. mul-1-neg N/A

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

       8. lower-fma.f64 N/A

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

       9. mul-1-neg N/A

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

       10. distribute-neg-frac2 N/A

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

       11. lower-/.f64 N/A

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

       12. distribute-neg-in N/A

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

       13. metadata-eval N/A

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

       14. unsub-neg N/A

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

       15. lower--.f64 88.0

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

   11. Applied rewrites 88.0%

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

Alternative 5: 82.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+216}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1950000000:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{-1 - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - a\right) - \frac{a}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.4e+216)
   (- x a)
   (if (<= z 1950000000.0) (fma a (/ y (- -1.0 t)) x) (- (- x a) (/ a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.4e+216) {
		tmp = x - a;
	} else if (z <= 1950000000.0) {
		tmp = fma(a, (y / (-1.0 - t)), x);
	} else {
		tmp = (x - a) - (a / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.4e+216)
		tmp = Float64(x - a);
	elseif (z <= 1950000000.0)
		tmp = fma(a, Float64(y / Float64(-1.0 - t)), x);
	else
		tmp = Float64(Float64(x - a) - Float64(a / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.4e+216], N[(x - a), $MachinePrecision], If[LessEqual[z, 1950000000.0], N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(x - a), $MachinePrecision] - N[(a / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.39999999999999991e216

   1. Initial program 87.3%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower--.f64 82.3

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

   5. Applied rewrites 82.3%

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

   if -1.39999999999999991e216 < z < 1.95e9

   1. Initial program 98.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower--.f64 47.9

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

   5. Applied rewrites 47.9%

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

   6. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{a} \]
   7. Step-by-step derivation

   8. Applied rewrites 5.5%

      \[\leadsto -a \]

   9. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x - \frac{a \cdot y}{1 + t}} \]
   10. Step-by-step derivation

       1. sub-neg N/A

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

       2. mul-1-neg N/A

          \[\leadsto x + \color{blue}{-1 \cdot \frac{a \cdot y}{1 + t}} \]

       3. +-commutative N/A

          \[\leadsto \color{blue}{-1 \cdot \frac{a \cdot y}{1 + t} + x} \]

       4. mul-1-neg N/A

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

       5. associate-/l* N/A

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

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

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

       7. mul-1-neg N/A

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

       8. lower-fma.f64 N/A

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

       9. mul-1-neg N/A

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

       10. distribute-neg-frac2 N/A

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

       11. lower-/.f64 N/A

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

       12. distribute-neg-in N/A

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

       13. metadata-eval N/A

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

       14. unsub-neg N/A

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

       15. lower--.f64 86.0

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

   11. Applied rewrites 86.0%

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

   if 1.95e9 < z

   1. Initial program 95.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{\frac{a \cdot z}{1 - z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.9%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 81.9%

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

Alternative 6: 71.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y}{z - 1}, x\right)\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+163}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{1 - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot a}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -1.2e+98)
   (fma a (/ y (- z 1.0)) x)
   (if (<= y 2.9e+163) (fma a (/ z (- 1.0 z)) x) (- x (/ (* y a) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.2e+98) {
		tmp = fma(a, (y / (z - 1.0)), x);
	} else if (y <= 2.9e+163) {
		tmp = fma(a, (z / (1.0 - z)), x);
	} else {
		tmp = x - ((y * a) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -1.2e+98)
		tmp = fma(a, Float64(y / Float64(z - 1.0)), x);
	elseif (y <= 2.9e+163)
		tmp = fma(a, Float64(z / Float64(1.0 - z)), x);
	else
		tmp = Float64(x - Float64(Float64(y * a) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -1.2e+98], N[(a * N[(y / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[y, 2.9e+163], N[(a * N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - N[(N[(y * a), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1999999999999999e98

   1. Initial program 94.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. lower-fma.f64 N/A

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

      8. lift-/.f64 N/A

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

      9. distribute-neg-frac N/A

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

      10. clear-num-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 94.9

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

   4. Applied rewrites 94.9%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 68.6

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

   7. Applied rewrites 68.6%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 65.1%

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

   if -1.1999999999999999e98 < y < 2.89999999999999998e163

   1. Initial program 97.9%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 85.6

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{\frac{a \cdot z}{1 - z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 82.4%

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

   if 2.89999999999999998e163 < y

   1. Initial program 97.0%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 65.3%

      \[\leadsto x - \frac{y \cdot a}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+14}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 15000:\\ \;\;\;\;\mathsf{fma}\left(-a, \mathsf{fma}\left(y - 1, z, y\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - a\right) - \frac{a}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.8e+14)
   (- x a)
   (if (<= z 15000.0) (fma (- a) (fma (- y 1.0) z y) x) (- (- x a) (/ a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.8e+14) {
		tmp = x - a;
	} else if (z <= 15000.0) {
		tmp = fma(-a, fma((y - 1.0), z, y), x);
	} else {
		tmp = (x - a) - (a / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.8e+14)
		tmp = Float64(x - a);
	elseif (z <= 15000.0)
		tmp = fma(Float64(-a), fma(Float64(y - 1.0), z, y), x);
	else
		tmp = Float64(Float64(x - a) - Float64(a / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.8e+14], N[(x - a), $MachinePrecision], If[LessEqual[z, 15000.0], N[((-a) * N[(N[(y - 1.0), $MachinePrecision] * z + y), $MachinePrecision] + x), $MachinePrecision], N[(N[(x - a), $MachinePrecision] - N[(a / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.8e14

   1. Initial program 94.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower--.f64 68.0

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

   5. Applied rewrites 68.0%

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

   if -3.8e14 < z < 15000

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 71.8

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

   5. Applied rewrites 71.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 71.6%

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

   if 15000 < z

   1. Initial program 95.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{\frac{a \cdot z}{1 - z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.9%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 81.9%

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

Alternative 8: 73.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+31}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 480000:\\ \;\;\;\;\mathsf{fma}\left(-a, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - a\right) - \frac{a}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.5e+31)
   (- x a)
   (if (<= z 480000.0) (fma (- a) y x) (- (- x a) (/ a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e+31) {
		tmp = x - a;
	} else if (z <= 480000.0) {
		tmp = fma(-a, y, x);
	} else {
		tmp = (x - a) - (a / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.5e+31)
		tmp = Float64(x - a);
	elseif (z <= 480000.0)
		tmp = fma(Float64(-a), y, x);
	else
		tmp = Float64(Float64(x - a) - Float64(a / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.5e+31], N[(x - a), $MachinePrecision], If[LessEqual[z, 480000.0], N[((-a) * y + x), $MachinePrecision], N[(N[(x - a), $MachinePrecision] - N[(a / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.50000000000000002e31

   1. Initial program 94.3%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower--.f64 66.8

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

   5. Applied rewrites 66.8%

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

   if -5.50000000000000002e31 < z < 4.8e5

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 67.2%

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

   if 4.8e5 < z

   1. Initial program 95.8%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{\frac{a \cdot z}{1 - z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 81.9%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 81.9%

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

Alternative 9: 97.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.2%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. distribute-neg-frac2 N/A

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

   6. div-inv N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. distribute-neg-frac N/A

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

   10. clear-num-rev N/A

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

   11. lower-/.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. distribute-neg-in N/A

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

   15. metadata-eval N/A

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

   16. unsub-neg N/A

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

   17. lower--.f64 97.3

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

4. Applied rewrites 97.3%

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

Alternative 10: 73.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+31}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 480000:\\ \;\;\;\;\mathsf{fma}\left(-a, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5.5e+31) (- x a) (if (<= z 480000.0) (fma (- a) y x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5.5e+31) {
		tmp = x - a;
	} else if (z <= 480000.0) {
		tmp = fma(-a, y, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5.5e+31)
		tmp = Float64(x - a);
	elseif (z <= 480000.0)
		tmp = fma(Float64(-a), y, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5.5e+31], N[(x - a), $MachinePrecision], If[LessEqual[z, 480000.0], N[((-a) * y + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.50000000000000002e31 or 4.8e5 < z

   1. Initial program 95.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower--.f64 74.9

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

   5. Applied rewrites 74.9%

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

   if -5.50000000000000002e31 < z < 4.8e5

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 67.2%

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

Alternative 11: 67.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{-15}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 2000000000:\\ \;\;\;\;\mathsf{fma}\left(a, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7e-15) (- x a) (if (<= z 2000000000.0) (fma a z x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7e-15) {
		tmp = x - a;
	} else if (z <= 2000000000.0) {
		tmp = fma(a, z, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7e-15)
		tmp = Float64(x - a);
	elseif (z <= 2000000000.0)
		tmp = fma(a, z, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7e-15], N[(x - a), $MachinePrecision], If[LessEqual[z, 2000000000.0], N[(a * z + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.0000000000000001e-15 or 2e9 < z

   1. Initial program 95.5%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower--.f64 75.2

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

   5. Applied rewrites 75.2%

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

   if -7.0000000000000001e-15 < z < 2e9

   1. Initial program 99.1%

      \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
   2. Add Preprocessing

   3. Taylor expanded in t around 0

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

      1. associate-/l* N/A

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

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

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

      3. +-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in y around 0

      \[\leadsto x + \color{blue}{\frac{a \cdot z}{1 - z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 61.6%

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

   9. Taylor expanded in z around 0

      \[\leadsto x + a \cdot \color{blue}{z} \]
   10. Step-by-step derivation

   11. Applied rewrites 61.5%

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

Alternative 12: 60.3% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

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 - a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - a), $MachinePrecision]

Derivation

1. Initial program 97.2%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

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

   1. lower--.f64 59.4

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

5. Applied rewrites 59.4%

   \[\leadsto \color{blue}{x - a} \]
6. Add Preprocessing

Alternative 13: 16.6% accurate, 11.7× speedup

Math

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

FPCore

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

C

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

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 = -a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := (-a)

Derivation

1. Initial program 97.2%

   \[x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

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

   1. lower--.f64 59.4

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

5. Applied rewrites 59.4%

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

6. Taylor expanded in x around 0

   \[\leadsto -1 \cdot \color{blue}{a} \]
7. Step-by-step derivation

8. Applied rewrites 14.7%

   \[\leadsto -a \]
9. Add Preprocessing

Developer Target 1: 99.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

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 - z) + 1.0d0)) * a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024298 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (* (/ (- y z) (+ (- t z) 1)) a)))

  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))