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

Percentage Accurate: 86.2% → 99.4%

Time: 8.9s

Alternatives: 13

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

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) / (a - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 86.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) / (a - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t\_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z}, y - z, x\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;t\_1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - y, t \cdot \frac{1}{z - a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- y z) t) (- a z))))
   (if (<= t_1 (- INFINITY))
     (fma (/ t (- a z)) (- y z) x)
     (if (<= t_1 5e+286) (+ t_1 x) (fma (- z y) (* t (/ 1.0 (- z a))) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y - z) * t) / (a - z);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma((t / (a - z)), (y - z), x);
	} else if (t_1 <= 5e+286) {
		tmp = t_1 + x;
	} else {
		tmp = fma((z - y), (t * (1.0 / (z - a))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y - z) * t) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(Float64(t / Float64(a - z)), Float64(y - z), x);
	elseif (t_1 <= 5e+286)
		tmp = Float64(t_1 + x);
	else
		tmp = fma(Float64(z - y), Float64(t * Float64(1.0 / Float64(z - a))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 5e+286], N[(t$95$1 + x), $MachinePrecision], N[(N[(z - y), $MachinePrecision] * N[(t * N[(1.0 / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 44.4%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   if -inf.0 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 5.0000000000000004e286

   1. Initial program 99.5%

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

   if 5.0000000000000004e286 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

   1. Initial program 35.3%

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

      1. +-commutative N/A

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

      2. frac-2neg N/A

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

      3. div-inv N/A

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

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

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

      5. associate-*l* N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. remove-double-neg N/A

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

      11. +-lowering-+.f64 N/A

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

      12. neg-lowering-neg.f64 N/A

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

      13. *-lowering-*.f64 N/A

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

      14. /-lowering-/.f64 N/A

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

      15. neg-sub0 N/A

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

      16. sub-neg N/A

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

      17. +-commutative N/A

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

      18. associate--r+ N/A

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

      19. neg-sub0 N/A

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

      20. remove-double-neg N/A

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

      21. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z}, y - z, x\right)\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot t}{a - z} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z - y, t \cdot \frac{1}{z - a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{a - z}, y - z, x\right)\\ t_2 := \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+286}:\\ \;\;\;\;t\_2 + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ t (- a z)) (- y z) x)) (t_2 (/ (* (- y z) t) (- a z))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 5e+286) (+ t_2 x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t / (a - z)), (y - z), x);
	double t_2 = ((y - z) * t) / (a - z);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 5e+286) {
		tmp = t_2 + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t / Float64(a - z)), Float64(y - z), x)
	t_2 = Float64(Float64(Float64(y - z) * t) / Float64(a - z))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 5e+286)
		tmp = Float64(t_2 + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 5e+286], N[(t$95$2 + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < -inf.0 or 5.0000000000000004e286 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

   1. Initial program 40.1%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 99.8

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

   4. Applied egg-rr 99.8%

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

   if -inf.0 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 5.0000000000000004e286

   1. Initial program 99.5%

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

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z}, y - z, x\right)\\ \mathbf{elif}\;\frac{\left(y - z\right) \cdot t}{a - z} \leq 5 \cdot 10^{+286}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot t}{a - z} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z}, y - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 55.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+133}:\\ \;\;\;\;t\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{+239}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* (- y z) t) (- a z))))
   (if (<= t_1 -1e+133) t (if (<= t_1 4e+239) x t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y - z) * t) / (a - z);
	double tmp;
	if (t_1 <= -1e+133) {
		tmp = t;
	} else if (t_1 <= 4e+239) {
		tmp = x;
	} else {
		tmp = t;
	}
	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) :: tmp
    t_1 = ((y - z) * t) / (a - z)
    if (t_1 <= (-1d+133)) then
        tmp = t
    else if (t_1 <= 4d+239) then
        tmp = x
    else
        tmp = t
    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 - z) * t) / (a - z);
	double tmp;
	if (t_1 <= -1e+133) {
		tmp = t;
	} else if (t_1 <= 4e+239) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((y - z) * t) / (a - z)
	tmp = 0
	if t_1 <= -1e+133:
		tmp = t
	elif t_1 <= 4e+239:
		tmp = x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y - z) * t) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -1e+133)
		tmp = t;
	elseif (t_1 <= 4e+239)
		tmp = x;
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((y - z) * t) / (a - z);
	tmp = 0.0;
	if (t_1 <= -1e+133)
		tmp = t;
	elseif (t_1 <= 4e+239)
		tmp = x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+133], t, If[LessEqual[t$95$1, 4e+239], x, t]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < -1e133 or 3.99999999999999996e239 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

   1. Initial program 53.7%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 46.7%

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

   6. Taylor expanded in x around 0

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

   8. Simplified 37.0%

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

   if -1e133 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 3.99999999999999996e239

   1. Initial program 99.4%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 73.5%

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

Alternative 4: 85.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -305000:\\ \;\;\;\;\mathsf{fma}\left(t, 1 - \frac{y}{z}, x\right)\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{-120}:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - z}, -t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -305000.0)
   (fma t (- 1.0 (/ y z)) x)
   (if (<= z 9.4e-120) (+ x (/ (* y t) (- a z))) (fma (/ z (- a z)) (- t) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -305000.0) {
		tmp = fma(t, (1.0 - (y / z)), x);
	} else if (z <= 9.4e-120) {
		tmp = x + ((y * t) / (a - z));
	} else {
		tmp = fma((z / (a - z)), -t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -305000.0)
		tmp = fma(t, Float64(1.0 - Float64(y / z)), x);
	elseif (z <= 9.4e-120)
		tmp = Float64(x + Float64(Float64(y * t) / Float64(a - z)));
	else
		tmp = fma(Float64(z / Float64(a - z)), Float64(-t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -305000.0], N[(t * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 9.4e-120], N[(x + N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] * (-t) + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -305000

   1. Initial program 66.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. neg-sub0 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

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

      16. /-lowering-/.f64 92.5

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

   5. Simplified 92.5%

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

   if -305000 < z < 9.40000000000000031e-120

   1. Initial program 94.4%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 89.4%

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

   if 9.40000000000000031e-120 < z

   1. Initial program 83.2%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 96.0

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

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. neg-lowering-neg.f64 84.1

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

   7. Simplified 84.1%

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

Alternative 5: 86.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, 1 - \frac{y}{z}, x\right)\\ \mathbf{if}\;z \leq -45000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.0022:\\ \;\;\;\;x + \frac{y \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma t (- 1.0 (/ y z)) x)))
   (if (<= z -45000.0) t_1 (if (<= z 0.0022) (+ x (/ (* y t) (- a z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(t, (1.0 - (y / z)), x);
	double tmp;
	if (z <= -45000.0) {
		tmp = t_1;
	} else if (z <= 0.0022) {
		tmp = x + ((y * t) / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(t, Float64(1.0 - Float64(y / z)), x)
	tmp = 0.0
	if (z <= -45000.0)
		tmp = t_1;
	elseif (z <= 0.0022)
		tmp = Float64(x + Float64(Float64(y * t) / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -45000.0], t$95$1, If[LessEqual[z, 0.0022], N[(x + N[(N[(y * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -45000 or 0.00220000000000000013 < z

   1. Initial program 72.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. neg-sub0 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

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

      16. /-lowering-/.f64 90.8

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

   5. Simplified 90.8%

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

   if -45000 < z < 0.00220000000000000013

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf

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

   5. Simplified 85.2%

      \[\leadsto x + \frac{\color{blue}{y} \cdot t}{a - z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 81.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{a}, y - z, x\right)\\ \mathbf{if}\;a \leq -1.7 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(t, 1 - \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ t a) (- y z) x)))
   (if (<= a -1.7e+109)
     t_1
     (if (<= a 1.6e+44) (fma t (- 1.0 (/ y z)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t / a), (y - z), x);
	double tmp;
	if (a <= -1.7e+109) {
		tmp = t_1;
	} else if (a <= 1.6e+44) {
		tmp = fma(t, (1.0 - (y / z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t / a), Float64(y - z), x)
	tmp = 0.0
	if (a <= -1.7e+109)
		tmp = t_1;
	elseif (a <= 1.6e+44)
		tmp = fma(t, Float64(1.0 - Float64(y / z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t / a), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.7e+109], t$95$1, If[LessEqual[a, 1.6e+44], N[(t * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.70000000000000003e109 or 1.60000000000000002e44 < a

   1. Initial program 82.4%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 97.8

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in a around inf

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

      1. /-lowering-/.f64 91.1

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

   7. Simplified 91.1%

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

   if -1.70000000000000003e109 < a < 1.60000000000000002e44

   1. Initial program 87.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. neg-sub0 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

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

      16. /-lowering-/.f64 82.6

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

   5. Simplified 82.6%

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

Alternative 7: 82.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{y - z}{a}, x\right)\\ \mathbf{if}\;a \leq -1.7 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 5.4 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(t, 1 - \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma t (/ (- y z) a) x)))
   (if (<= a -1.7e+109)
     t_1
     (if (<= a 5.4e+44) (fma t (- 1.0 (/ y z)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(t, ((y - z) / a), x);
	double tmp;
	if (a <= -1.7e+109) {
		tmp = t_1;
	} else if (a <= 5.4e+44) {
		tmp = fma(t, (1.0 - (y / z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(t, Float64(Float64(y - z) / a), x)
	tmp = 0.0
	if (a <= -1.7e+109)
		tmp = t_1;
	elseif (a <= 5.4e+44)
		tmp = fma(t, Float64(1.0 - Float64(y / z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.7e+109], t$95$1, If[LessEqual[a, 5.4e+44], N[(t * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.70000000000000003e109 or 5.4e44 < a

   1. Initial program 82.4%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 89.1

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

   5. Simplified 89.1%

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

   if -1.70000000000000003e109 < a < 5.4e44

   1. Initial program 87.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. neg-sub0 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

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

      16. /-lowering-/.f64 82.6

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

   5. Simplified 82.6%

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

Alternative 8: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{if}\;a \leq -1.6 \cdot 10^{+109}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(t, 1 - \frac{y}{z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ t a) x)))
   (if (<= a -1.6e+109) t_1 (if (<= a 4e+44) (fma t (- 1.0 (/ y z)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (t / a), x);
	double tmp;
	if (a <= -1.6e+109) {
		tmp = t_1;
	} else if (a <= 4e+44) {
		tmp = fma(t, (1.0 - (y / z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(t / a), x)
	tmp = 0.0
	if (a <= -1.6e+109)
		tmp = t_1;
	elseif (a <= 4e+44)
		tmp = fma(t, Float64(1.0 - Float64(y / z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.6e+109], t$95$1, If[LessEqual[a, 4e+44], N[(t * N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.6000000000000001e109 or 4.0000000000000004e44 < a

   1. Initial program 82.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 86.9

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

   5. Simplified 86.9%

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

   if -1.6000000000000001e109 < a < 4.0000000000000004e44

   1. Initial program 87.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. neg-sub0 N/A

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

      7. div-sub N/A

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

      8. *-inverses N/A

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

      9. associate-+l- N/A

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

      10. neg-sub0 N/A

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

      11. mul-1-neg N/A

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

      12. +-commutative N/A

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

      13. mul-1-neg N/A

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

      14. unsub-neg N/A

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

      15. --lowering--.f64 N/A

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

      16. /-lowering-/.f64 82.6

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

   5. Simplified 82.6%

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

Alternative 9: 74.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -300000:\\ \;\;\;\;x + \mathsf{fma}\left(t, \frac{a}{z}, t\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -300000.0)
   (+ x (fma t (/ a z) t))
   (if (<= z 3.8e-112) (fma y (/ t a) x) (+ t x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -300000.0) {
		tmp = x + fma(t, (a / z), t);
	} else if (z <= 3.8e-112) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -300000.0)
		tmp = Float64(x + fma(t, Float64(a / z), t));
	elseif (z <= 3.8e-112)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(t + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -300000.0], N[(x + N[(t * N[(a / z), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e-112], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(t + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3e5

   1. Initial program 66.8%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 91.5

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

   4. Applied egg-rr 91.5%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

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

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

      6. mul-1-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. /-lowering-/.f64 N/A

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

      9. --lowering--.f64 N/A

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

      10. mul-1-neg N/A

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

      11. neg-lowering-neg.f64 89.3

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

   7. Simplified 89.3%

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

   8. Taylor expanded in z around inf

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

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. associate-+l+ N/A

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

      4. +-lowering-+.f64 N/A

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

      5. *-rgt-identity N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. distribute-lft-out N/A

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

      9. +-commutative N/A

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

      10. distribute-lft-out N/A

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

      11. *-rgt-identity N/A

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

      12. accelerator-lowering-fma.f64 N/A

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

      13. /-lowering-/.f64 84.1

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

   10. Simplified 84.1%

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

   if -3e5 < z < 3.79999999999999995e-112

   1. Initial program 93.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 80.5

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

   5. Simplified 80.5%

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

   if 3.79999999999999995e-112 < z

   1. Initial program 84.7%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 74.4%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -300000:\\ \;\;\;\;x + \mathsf{fma}\left(t, \frac{a}{z}, t\right)\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 75.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5200:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5200.0) (+ t x) (if (<= z 3.8e-112) (fma y (/ t a) x) (+ t x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5200.0) {
		tmp = t + x;
	} else if (z <= 3.8e-112) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5200.0)
		tmp = Float64(t + x);
	elseif (z <= 3.8e-112)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(t + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5200.0], N[(t + x), $MachinePrecision], If[LessEqual[z, 3.8e-112], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(t + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5200 or 3.79999999999999995e-112 < z

   1. Initial program 78.2%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 77.8%

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

   if -5200 < z < 3.79999999999999995e-112

   1. Initial program 93.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 80.5

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

   5. Simplified 80.5%

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

4. Final simplification 79.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5200:\\ \;\;\;\;t + x\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 96.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. /-lowering-/.f64 N/A

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

   6. --lowering--.f64 N/A

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

   7. --lowering--.f64 95.5

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

4. Applied egg-rr 95.5%

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

Alternative 12: 62.1% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 5.3 \cdot 10^{+188}:\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (if (<= a 5.3e+188) (+ t x) x))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 5.3e+188) {
		tmp = t + x;
	} else {
		tmp = x;
	}
	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 <= 5.3d+188) then
        tmp = t + x
    else
        tmp = x
    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 <= 5.3e+188) {
		tmp = t + x;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= 5.3e+188:
		tmp = t + x
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= 5.3e+188)
		tmp = Float64(t + x);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= 5.3e+188)
		tmp = t + x;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, 5.3e+188], N[(t + x), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if a < 5.29999999999999988e188

   1. Initial program 86.1%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 67.3%

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

   if 5.29999999999999988e188 < a

   1. Initial program 76.1%

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

   3. Taylor expanded in x around inf

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

   5. Simplified 84.1%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 5.3 \cdot 10^{+188}:\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 18.6% accurate, 26.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.3%

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

3. Taylor expanded in z around inf

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

5. Simplified 65.9%

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

6. Taylor expanded in x around 0

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

8. Simplified 20.2%

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

Developer Target 1: 99.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - z}{a - z} \cdot t\\ \mathbf{if}\;t < -1.0682974490174067 \cdot 10^{-39}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t < 3.9110949887586375 \cdot 10^{-141}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y z) (- a z)) t))))
   (if (< t -1.0682974490174067e-39)
     t_1
     (if (< t 3.9110949887586375e-141) (+ x (/ (* (- y z) t) (- a z))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - z) / (a - z)) * t);
	double tmp;
	if (t < -1.0682974490174067e-39) {
		tmp = t_1;
	} else if (t < 3.9110949887586375e-141) {
		tmp = x + (((y - z) * t) / (a - z));
	} 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) :: tmp
    t_1 = x + (((y - z) / (a - z)) * t)
    if (t < (-1.0682974490174067d-39)) then
        tmp = t_1
    else if (t < 3.9110949887586375d-141) then
        tmp = x + (((y - z) * t) / (a - z))
    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 = x + (((y - z) / (a - z)) * t);
	double tmp;
	if (t < -1.0682974490174067e-39) {
		tmp = t_1;
	} else if (t < 3.9110949887586375e-141) {
		tmp = x + (((y - z) * t) / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - z) / (a - z)) * t)
	tmp = 0
	if t < -1.0682974490174067e-39:
		tmp = t_1
	elif t < 3.9110949887586375e-141:
		tmp = x + (((y - z) * t) / (a - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - z) / Float64(a - z)) * t))
	tmp = 0.0
	if (t < -1.0682974490174067e-39)
		tmp = t_1;
	elseif (t < 3.9110949887586375e-141)
		tmp = Float64(x + Float64(Float64(Float64(y - z) * t) / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - z) / (a - z)) * t);
	tmp = 0.0;
	if (t < -1.0682974490174067e-39)
		tmp = t_1;
	elseif (t < 3.9110949887586375e-141)
		tmp = x + (((y - z) * t) / (a - z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]}, If[Less[t, -1.0682974490174067e-39], t$95$1, If[Less[t, 3.9110949887586375e-141], N[(x + N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024198 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< t -10682974490174067/10000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y z) (- a z)) t)) (if (< t 312887599100691/80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (/ (* (- y z) t) (- a z))) (+ x (* (/ (- y z) (- a z)) t)))))

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